Bocconi Mathematics 30062 · exam prep

Bocconi Mathematics 30062:
every formula, one search away.

585 entries from the Bocconi Mathematics 30062 Module 1 (General) lectures — every definition, formula, theorem and proof — each with a plain-English explanation and a breakdown of what every symbol stands for. Built to help you prepare efficiently and drill until it sticks.

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Sets & set operations35

What it means

A set is a collection of objects or elements. Sets are usually denoted with capital letters and elements with lowercase letters.

Components

$A, B$ sets, denoted by capital letters
$x, a$ elements, denoted by lowercase letters

What it means

A set can be described either by enumerating all its elements, or by giving a rule that every element must satisfy. Infinite sets typically require the rule-based description.

Components

${\cdot}$ set notation
$x ∣ P (x)$ the set of x satisfying property P

What it means

x ∈ A denotes that x is an element of A, while x ∉ A denotes that x is not an element of A.

Components

$\in$ is an element of
$\in /$ is not an element of

What it means

A is a subset of B if every element of A is also an element of B. This allows the possibility that A = B.

Components

$\subseteq$ subset or equal
$A, B$ sets being compared

What it means

A is a proper subset of B if A is a subset of B, but B contains at least one element that A does not — so A and B cannot be equal.

Components

$\subset$ proper subset
$\exists$ there exists

What it means

A is not a subset of B if some element of A fails to belong to B.

Components

$⊈$ is not a subset of

What it means

Two sets are equal exactly when they contain the same elements, which can be shown by proving each is a subset of the other (double inclusion).

Components

$A = B$ set equality

What it means

The empty set is the unique set that contains no elements at all.

Components

$\emptyset$ the empty set

What it means

The union of A and B contains every element that is in A, in B, or in both.

Components

$\cup$ union operator

What it means

The intersection of A and B contains only elements that belong to both A and B.

Components

$\cap$ intersection operator

What it means

Two sets are disjoint if they share no common elements, meaning membership in one implies non-membership in the other; their intersection is the empty set.

Components

$A \cap B = \emptyset$ condition for disjointness

What it means

Union and intersection generalize naturally to more than two sets, combining or intersecting an entire indexed family of sets.

Components

$⋃_{i = 1}^{N} A_{i}$ union over the family of sets A_1,...,A_N
$⋂_{i = 1}^{N} A_{i}$ intersection over the family of sets A_1,...,A_N

What it means

The set difference A \ B contains the elements of A that are not in B. B need not be a subset of A, and A \ A = ∅.

Components

$∖$ set difference operator

What it means

The order of the sets does not matter when taking a union or intersection, just as 4+5 = 5+4 for numbers.

Components

$\cup, \cap$ union and intersection

What it means

Grouping does not matter when taking repeated unions or intersections, analogous to 4*(5*6) = (4*5)*6.

Components

$\cup, \cap$ union and intersection

What it means

Union distributes over intersection and intersection distributes over union, mirroring the numeric distributive law 4*(5+6) = 4*5 + 4*6.

Components

$\cup, \cap$ union and intersection

What it means

The universal set U is the larger ambient set within which all other sets under discussion are considered subsets, e.g. all cities when discussing Italian cities.

Components

$U$ the universal set

What it means

The complement of A (relative to universal set U) consists of all elements of U that are not in A.

Components

$A^{c}$ complement of A
$U$ universal set

What it means

Removing B from A is the same as intersecting A with the complement of B.

Components

$A ∖ B$ set difference
$B^{c}$ complement of B

What it means

Taking the complement of the complement of a set returns the original set. Assigned as homework to prove using the universal set U with A ⊆ U.

Components

$(A^{c})^{c}$ complement of the complement of A

What it means

The complement of an intersection is the union of the complements, and the complement of a union is the intersection of the complements. These laws describe how complementing interacts with union and intersection.

Components

$(A \cap B)^{c}$ complement of the intersection
$A^{c} \cup B^{c}$ union of the complements

What it means

The proof shows both inclusions at once by chaining logical equivalences: an element fails to be in both A and B exactly when it fails to be in at least one of them, which is the definition of being in the union of the complements.

What it means

The natural numbers, or counting numbers, are closed under addition and multiplication but not under subtraction or division (e.g. 1-2 ∉ ℕ, 3/4 ∉ ℕ).

Components

$N$ set of natural numbers

What it means

The integers extend the natural numbers to include negatives, and are closed under addition, subtraction, and multiplication, but not division.

Components

$Z$ set of integers

What it means

The rational numbers are ratios of integers, and are closed under addition, subtraction, multiplication, and division (by nonzero elements).

Components

$Q$ set of rational numbers

What it means

This classic proof by contradiction shows that √2 cannot be written as a ratio of coprime integers, since assuming it can forces both the numerator and denominator to be even, contradicting their coprimality.

What it means

The real numbers consist of the rational numbers together with all the irrational numbers (numbers not expressible as a ratio of integers), and are closed under addition, subtraction, multiplication, and division.

Components

$R$ set of real numbers
$Q^{c}$ irrational numbers

What it means

Each number system is a proper subset of the next: naturals sit inside integers, inside rationals, inside the reals.

Components

$\subset$ proper subset

What it means

Q is dense in R (between any two reals lies a rational) but not continuous (it has 'holes' like √2). R is both dense and continuous — it has no holes.

Components

$p, q, r$ real/rational numbers

What it means

The real numbers have a total order: for any two reals exactly one of >, <, = holds, and ≥ (or ≤) is a total ordering on R.

Components

$>, <, =$ order relations
$\geq, \leq$ weak order relations

What it means

Unlike R, the vectors of R² cannot be totally ordered by ≥ in a natural componentwise sense — neither (1,3) nor (3,1) dominates the other.

What it means

A set of real numbers is an interval if it contains every point between any two of its elements — there are no gaps.

Components

$A$ subset of R
${z ∣ x \leq z \leq y}$ all points between x and y

What it means

Closed intervals include both endpoints, open intervals include neither, and half-open intervals include exactly one endpoint.

Components

$[x, y]$ closed interval
$(x, y)$ open interval
$[x, y), (x, y]$ half-open intervals

What it means

Intervals can extend infinitely in one or both directions; the whole real line itself is the interval (-∞, ∞).

Components

$\infty, - \infty$ positive/negative infinity, used as unbounded endpoints

What it means

The extended reals add the two points +∞ and -∞ to R, forming a closed interval that includes infinity at both ends, unlike R itself which is only (-∞, ∞).

Components

$R$ extended real line

Real numbers & completeness16

What it means

For a nonempty set A ⊆ R, an upper bound is any real number at least as large as every element of A, and a lower bound is any real number at most as small as every element of A.

Components

$h$ candidate bound
$A$ nonempty subset of R

What it means

A set is bounded above if it has some upper bound, bounded below if it has some lower bound, and bounded if it has both. A set lacking one of these is called unbounded (above or below).

Components

$A$ subset of R

What it means

The maximum (minimum) of A is an element of A that is greater (less) than or equal to every other element. Unlike bounds, max/min must belong to A itself, and may fail to exist even if A is bounded, e.g. (0,1) has neither.

Components

$max A, min A$ maximum and minimum of A

What it means

If a set has a maximum or minimum at all, that extremal element is unique. Assigned as homework to prove.

Components

$A$ nonempty subset of R

What it means

The supremum is the least upper bound and the infimum is the greatest lower bound of A. Unlike max/min, sup and inf need not belong to A, but they always exist when A is bounded above/below respectively (by completeness).

Components

$sup A$ least upper bound of A
$in f A$ greatest lower bound of A

What it means

Whenever a maximum exists it coincides with the supremum, and conversely, if the supremum happens to belong to the set, it is also the maximum. The analogous statements hold for minimum and infimum.

Components

$sup A, max A$ supremum and maximum of A
$in f A, min A$ infimum and minimum of A

What it means

Every nonempty subset of R that is bounded above has a supremum, and every nonempty subset bounded below has an infimum. This is the completeness principle: R has no 'holes' in its number line, unlike Q.

Components

$A$ nonempty subset of R

What it means

The absolute value of x measures its distance from 0 on the number line, regardless of sign.

Components

$∣ x ∣$ absolute value of x

What it means

Absolute value is always non-negative, is zero only at zero, is multiplicative, and satisfies the triangle inequality, which bounds the size of a sum by the sum of sizes.

Components

$∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ triangle inequality

What it means

The set of x within absolute value c of zero forms the open interval (-c, c); by contrast, {x : |x|>c} does not form a single interval since it splits into two pieces.

Components

$c$ positive constant

What it means

The distance between two real numbers is the absolute value of their difference; the distance from 0 to x is simply |x|.

Components

$d (x, y)$ distance between x and y

What it means

A neighborhood of x₀ with radius ε is the open, bounded interval centered at x₀ with half-width ε, also called the open ball around x₀.

Components

$B_{ϵ} (x_{0})$ neighborhood of x_0 with radius epsilon
$ϵ$ radius
$x_{0}$ center point

What it means

A right neighborhood extends from x₀ up to (but not including) x₀+ε and includes x₀; a left neighborhood extends down to x₀-ε (excluded) up to and including x₀.

Components

$B_{ϵ}^{+} (x_{0})$ right half-neighborhood
$B_{ϵ}^{-} (x_{0})$ left half-neighborhood

What it means

An infinite neighborhood extends without bound in one direction from x₀ and is always an open interval.

Components

$B_{x_{0}} (\infty), B_{x_{0}} (- \infty)$ one-sided infinite neighborhoods

What it means

x is the supremum of A exactly when x is an upper bound of A and every left-neighborhood of x, however small, still contains some point of A — i.e., no smaller number can be an upper bound. Assigned as homework to prove.

Components

$B_{ϵ}^{-} (x)$ left neighborhood of x with radius epsilon

What it means

x is the infimum of A exactly when x is a lower bound of A and every right-neighborhood of x, however small, still contains some point of A — i.e., no larger number can be a lower bound. Assigned as homework to prove.

Components

$B_{ϵ}^{+} (x)$ right neighborhood of x with radius epsilon

Vectors & Rⁿ34

What it means

The Cartesian product of two sets pairs each element of the first set with each element of the second, forming a new set of ordered pairs.

Components

$\times$ Cartesian product operator
$(a_{1}, a_{2})$ ordered pair

What it means

R² is the set of all ordered pairs of real numbers, each representing a unique point in the Cartesian plane.

Components

$R^{2}$ the Cartesian plane
$(x, y)$ ordered pair / point

What it means

Each ordered pair in R² can also be interpreted as a vector, drawn as an arrow from the origin (0,0) to that point.

Components

$(12)$ column vector representation of the point (1,2)

What it means

R³ is the set of ordered triples of real numbers, each representing a point in three-dimensional space; vectors in R³ are arrows from the origin to that point.

Components

$R^{3}$ three-dimensional Cartesian space
$(x, y, z)$ ordered triple

What it means

Rⁿ generalizes the plane and 3-space to n dimensions: it is the set of all ordered n-tuples of real numbers, representable as either column or row vectors.

Components

$R^{n}$ n-dimensional Cartesian space
$(x_{1}, \dots, x_{n})$ ordered n-tuple

What it means

A vector in Rⁿ can be written either as a vertical column of its components or as a horizontal row; both notations represent the same object.

Components

$x_{1}, \dots, x_{n}$ components of the vector

What it means

Two vectors of the same dimension are added component by component, producing a resultant vector.

Components

$x, y \in R^{n}$ vectors of the same dimension n
$x_{i}, y_{i}$ i-th components of x and y

What it means

Vector addition is only defined when the two vectors live in the same space Rⁿ; adding vectors of different lengths makes no sense.

What it means

Geometrically, placing u and v tail-to-tail and completing a parallelogram shows their sum as the interior diagonal.

Components

$u, v$ two vectors sharing a starting point

What it means

Vector addition on Rⁿ satisfies the same four basic algebraic properties as ordinary addition of real numbers.

Components

$x, y, z \in R^{n}$ arbitrary vectors
$0$ zero vector
$- x$ additive inverse of x

What it means

Multiplying a vector by a scalar scales every component by that same factor, stretching/shrinking (and possibly flipping) the vector.

Components

$α \in R$ scalar
$x \in R^{n}$ vector

What it means

Scalar multiplication distributes over vector addition and over sums of scalars, has a multiplicative identity, and is associative.

Components

$x, y \in R^{n}$ vectors
$α, β \in R$ scalars

What it means

Rⁿ, together with vector addition and scalar multiplication, forms a vector space because it satisfies all eight defining algebraic properties.

What it means

A linear combination combines scalar multiplication and addition of any finite collection of vectors, and Rⁿ is closed under this operation.

Components

$α_{i}$ scalar coefficients
$x_{i}$ vectors being combined

What it means

The inner product multiplies vectors component-wise and sums the results, producing a single scalar that reflects how aligned two vectors are.

Components

$x, y \in R^{n}$ vectors
$x \cdot y$ resulting scalar

What it means

The sign of the dot product tells us the general directional relationship between two vectors.

What it means

Two vectors are orthogonal if their dot product is zero; a set of vectors is orthogonal if every pair in the set is orthogonal.

Components

$x ⊥ y$ x is orthogonal to y

What it means

The dot product is commutative, distributes over addition, is compatible with scalar multiplication, and is non-negative, vanishing only for the zero vector.

Components

$x, y, z \in R^{n}$ vectors
$α \in R$ scalar

What it means

Two vectors are equal exactly when all corresponding components match.

What it means

x is weakly greater than y if every component of x is at least the corresponding component of y.

What it means

x is strictly greater than y if x is weakly greater in every component and strictly greater in at least one component.

What it means

x is strongly greater than y if every single component of x strictly exceeds the corresponding component of y.

What it means

Two vectors are not comparable when neither is weakly greater than the other, i.e., one exceeds the other in some coordinates but not in others.

What it means

The natural order on Rⁿ is only a partial order (not total, unlike on R): strong inequality implies strict, which implies weak, but not conversely.

What it means

The order relations applied to the zero vector define three grades of positivity for a vector.

What it means

In R, the norm of a number is its absolute value, which also equals its distance from zero.

What it means

The Euclidean norm generalizes absolute value/Pythagorean length to n dimensions, giving the length of a vector as the square root of the sum of squared components.

Components

$x \in R^{n}$ vector
$∥ x ∥$ length (norm) of x

What it means

A unit vector has length 1; any nonzero vector can be normalized by dividing by its norm; a set of pairwise orthogonal unit vectors is called orthonormal.

What it means

The norm behaves like a length measure: it is non-negative, scales proportionally, satisfies the triangle inequality, and the dot product is bounded by the product of norms (Cauchy-Schwarz).

Components

$x, y \in R^{n}$ vectors
$α \in R$ scalar

What it means

If two vectors are orthogonal, the squared length of their sum equals the sum of their squared lengths — the vector generalization of the classic Pythagorean theorem.

Components

$x ⊥ y$ x and y orthogonal

What it means

Expanding the norm of the sum via the inner product and using orthogonality (x·y=0) removes the cross term, leaving the Pythagorean identity.

What it means

The sum of the lengths of two vectors is always at least the length of their sum, mirroring the fact that any side of a triangle is no longer than the sum of the other two.

What it means

The distance between two vectors is defined as the norm of their difference, generalizing the ordinary Euclidean distance formula to n dimensions.

Components

$x, y \in R^{n}$ points/vectors
$d (x, y)$ Euclidean distance between x and y

What it means

Distance is non-negative (zero only for identical points), symmetric, and satisfies the triangle inequality, making it a valid metric on Rⁿ.

Components

$x, y, z \in R^{n}$ points

Convexity in Rⁿ5

What it means

An interval in R has the property that all numbers between any two of its points also belong to the interval; this is the seed idea for convexity.

What it means

A set in the plane or space is convex if it contains the entire straight segment between any two of its points; a circle or square is convex, but a star shape is not.

What it means

A convex combination of two vectors is a weighted average of them with weights summing to 1; letting α range over [0,1] traces out the line segment joining x and y.

Components

$x, y \in R^{n}$ endpoints
$α \in [0, 1]$ weight parameter

What it means

A convex combination generalizes to any finite number of points: it is a linear combination whose coefficients are non-negative and sum to one.

Components

$α_{i} \geq 0$ non-negative weights
$\sum α_{i} = 1$ weights sum to one

What it means

A subset of Rⁿ is convex if it contains the entire line segment between any two of its points — extending the interval concept from R to Rⁿ. Examples: triangles, spheres, cubes; non-examples: stars, tori, disjoint unions of intervals.

Components

$A \subseteq R^{n}$ the set in question

Topology of Rⁿ46

What it means

A neighborhood of a point x0 in R is an open interval centered at x0 with radius ε, equivalently the set of points within distance ε of x0.

Components

$x_{0}$ center point
$ϵ > 0$ radius
$B_{ϵ} (x_{0})$ neighborhood (ball) of radius ε around x0

What it means

The neighborhood concept extends to Rⁿ using the Euclidean distance: it is the open ball of radius ε centered at x0 (an interval in R, a disk in R², a solid sphere in R³, etc.).

Components

$x_{0} \in R^{n}$ center point
$ϵ > 0$ radius
$d (x, x_{0})$ Euclidean distance from x to x0

What it means

A point of A is interior if some entire small ball around it is still contained inside A. The set of all such points is denoted intA.

Components

$A \subseteq R^{n}$ the set
$int A$ set of interior points of A

What it means

Illustrates that endpoints/boundary values of closed or half-open sets are excluded from the interior, since no small ball around them stays inside the set.

What it means

A point not in A is exterior to A if it has a neighborhood entirely contained in the complement of A. The set of all exterior points is denoted extA.

Components

$A^{c}$ complement of A
$ext A$ set of exterior points of A

What it means

The exterior of an interval or ball is everything strictly outside it, regardless of whether the boundary itself belongs to the set.

What it means

A boundary point is neither interior nor exterior to A: every neighborhood of it meets both A and its complement. The set of boundary points is written ∂A.

Components

$\partial A$ set of boundary points of A

What it means

The boundary of intervals, balls, and rectangles consists exactly of their 'edge' points, whether or not those points are actually included in the set.

What it means

A set and its complement share exactly the same boundary, and the closure of a set is obtained by adding its boundary points to the set itself.

Components

$A = cl A$ closure of A

What it means

Because both rationals and irrationals are dense in R, every real number's neighborhood contains both, so every point of R is a boundary point of Q.

What it means

Splits into the case where the test point is rational and where it is irrational, using density of both Q and its complement to always find one of each type inside any neighborhood.

What it means

A point of A is isolated if it has a neighborhood containing no other points of A besides itself.

What it means

An open interval has no isolated points since every point has neighbors nearby, whereas an added lone point (like 6) or a discrete set like the integers consists entirely of isolated points.

What it means

Since every neighborhood of an isolated point contains points outside A as well as the point itself, isolated points always qualify as boundary points.

What it means

A point is a limit point of A if every neighborhood of it, no matter how small, contains some point of A other than itself. Equivalently there is some y in A with 0 < d(x,y) < ε for every ε. The set of all limit points is the derived set A′.

Components

$A^{'}$ derived set of A (set of all limit points)

What it means

For an interval, the derived set adds the endpoints; for an open exterior region, it adds the boundary sphere; a line is a closed set equal to its own derived set.

What it means

Summarizes how interior, boundary, isolated, and limit points relate to one another: interior points are always limit points, isolated points are boundary but never limit points, and limit points always have infinitely many nearby points of A.

What it means

The general neighborhood in Rⁿ is also called an open ball; in R it looks like an interval, in R² a disk, and in R³ a solid sphere.

What it means

Before defining open and closed sets, we classify how any point in Rⁿ relates to a given set A using these five point types.

What it means

The exterior of the half-open rectangle consists of all points strictly outside the rectangle's range in either coordinate.

What it means

Because an isolated point has a neighborhood containing no other points of A, it cannot satisfy the limit point condition, which requires nearby points of A distinct from x.

What it means

Vectors may be written in bold, with an arrow hat, or simply by stating the set they belong to; going forward the plain notation x ∈ Rⁿ is used.

What it means

A set is open if every one of its points is an interior point, i.e. around each point there is a small ball entirely contained in the set.

Components

$int A$ interior of A
$B_{ϵ} (x)$ open ball of radius ϵ around x

What it means

Open intervals (a,b) are open sets, while closed intervals [a,b] and half-open intervals like [a,b) are not.

What it means

Combining open sets by union preserves openness, as illustrated by (0,1) ∪ (3,4) being open.

What it means

Sets defined with ≤ or ≥ typically include their boundary, so they are not open (boundary points are not interior points).

What it means

Sets defined with strict inequalities < or > exclude their boundary, so every point is interior and the set is open.

What it means

A set is closed if it contains all of its own boundary points.

Components

$\partial A$ boundary of A: points where every ball meets both A and its complement

What it means

Closed intervals contain their endpoints (their boundary), so they are closed sets; open and half-open intervals are not.

What it means

A finite union of closed sets is always closed.

What it means

Unlike finite unions, an infinite union of closed sets can fail to be closed — here the union of shrinking closed intervals equals the open interval (-1,1).

What it means

Sets defined with ≤ or ≥ generally are closed because they include their boundary; sets defined with strict inequalities are generally not closed.

What it means

Whether a set is open or closed is relative to the universal set (ambient space) it sits inside, not an absolute property of the set alone.

What it means

A set that is both open and closed is called clopen; in ℝ the only clopen sets are ℝ itself and the empty set.

What it means

A set is open exactly when its complement is closed, giving a duality between the two notions and a tool to prove one from the other.

Components

$A^{c}$ complement of A

What it means

Example applying the open/closed complement theorem: showing A^c is an open interval proves A is closed without directly checking A's boundary.

What it means

Another application of the complement theorem: since the closed disk is closed, its complement (points strictly outside the unit circle) must be open.

What it means

A set is closed precisely when every limit point of the set (a point every neighborhood of which contains other points of the set) already belongs to the set.

What it means

A set is bounded if it fits entirely inside some ball around the origin, i.e. it does not extend infinitely in any direction.

Components

$K$ a fixed positive radius bounding all points of A
$B_{K} (0)$ ball of radius K centered at the origin containing A

What it means

This set is contained inside a ball of radius 2, so it is bounded by definition.

What it means

This set (the region above a parabola) extends infinitely, so no ball around the origin can contain it — it is unbounded.

What it means

Compact sets are those that are both closed and bounded; they behave much like finite sets and guarantee that continuous functions on them attain a maximum.

What it means

One of the most important consequences of compactness: a continuous function defined on a compact set always achieves a maximum value; this need not hold otherwise.

What it means

Table of worked examples showing that a set must satisfy both closedness and boundedness to be compact — failing either property rules out compactness.

What it means

Every open ball (neighborhood) in ℝⁿ is itself an open set: around any point inside the ball, a smaller ball still fits inside it.

Components

$B_{ϵ} (x)$ open ball (neighborhood) of radius ϵ centered at x

What it means

To show a ball is open, pick any point y inside it and construct a smaller ball around y (using the leftover distance to the boundary) that still lies within the original ball; this makes every point of the ball interior.

Components

$δ$ radius of the smaller ball around y, chosen as ϵ − ‖x−y‖

Functions: basics36

What it means

A function is a rule that assigns to every element of a domain set A precisely one element of a codomain set B.

Components

$A$ domain of f
$B$ codomain of f
$f (x) = y$ f associates y ∈ B to x ∈ A

What it means

Standard vocabulary: each x has exactly one image y=f(x), but a given y may have zero, one, or many preimages in A.

Components

$x$ independent variable / input
$y = f (x)$ dependent variable / output, the image of x

What it means

When a formula for f is not defined everywhere, the natural domain is the largest set of inputs for which the formula makes sense (e.g. excluding zero denominators, negative numbers under even roots, non-positive log arguments).

What it means

Worked examples of finding the natural domain: avoid division by zero, require non-negative arguments under even roots, and require positive arguments for logarithms.

What it means

The range (or image) of f is the set of all actual output values, which is always a subset of the codomain but need not equal it.

Components

$Im (f)$ range / image of f

What it means

A function is surjective (onto) if every element of the codomain is actually attained as an output, i.e. the range equals the whole codomain.

What it means

f(x)=x⁴ onto ℝ is not surjective because only non-negative values are produced; restricting the codomain to ℝ≥0 makes it surjective.

What it means

A function is injective (one-to-one) if distinct inputs always give distinct outputs — no two different domain points map to the same image.

What it means

f(x)=x² fails injectivity on all of ℝ since two different inputs can share an output, but restricting the domain to non-negative numbers restores injectivity.

What it means

A bijective function is both one-to-one and onto, meaning every element of B corresponds to exactly one element of A.

What it means

The cubing function on ℝ is surjective (every real is a cube) and injective (cube roots are unique), hence bijective.

What it means

The course generalizes function concepts to cover these four types: scalar functions of one or several variables, and vector-valued functions of one or several variables.

What it means

The graph of a function is the set of all input-output pairs, viewed as a subset of the product of domain and codomain (e.g. a curve in ℝ² for f:ℝ→ℝ).

Components

$Gr (f)$ graph of f

What it means

For f:ℝ→ℝ, the graph must pass the vertical line test since each x has exactly one image y.

What it means

For functions of two real variables, the graph is a two-dimensional surface embedded in three-dimensional space.

What it means

A linear function has a constant rate of change m (the slope) and vertical intercept b.

Components

$m$ slope
$b$ y-intercept

What it means

A quadratic function produces a parabola, with shape controlled by the leading coefficient a.

Components

$a, b, c$ real coefficients, a ≠ 0

What it means

An inverse proportion function describes a hyperbolic relationship where y decreases as x increases, undefined at x=0.

Components

$k$ proportionality constant

What it means

A power function raises x to a fixed exponent α; its shape and symmetry (even, odd, or neither) depend on the rational value of α.

Components

$α$ exponent, α ∈ ℚ

What it means

An exponential function has a constant base raised to a variable exponent, giving rapid growth (α>1) or decay (0<α<1).

Components

$α$ base, α > 0, α ≠ 1

What it means

The logarithmic function is the inverse of the exponential function with the same base, defined only for positive x.

Components

$α$ base of the logarithm

What it means

The standard periodic trigonometric functions sine, cosine and tangent.

What it means

A piecewise function is defined by different formulas on different parts of its domain.

What it means

Adding a constant a to a function shifts its graph vertically up by a units (down if a is negative).

Components

$a$ vertical shift amount

What it means

Replacing x with x+a shifts the graph of f horizontally to the left by a units (right if a is negative).

Components

$a$ horizontal shift amount

What it means

Negating a function reflects its graph vertically across the x-axis.

What it means

Negating the input to a function reflects its graph horizontally across the y-axis.

What it means

There are two absolute-value transformations of a function: applying |·| to the output flips negative parts up, while applying |·| to the input mirrors the right half of the graph onto the left.

What it means

Two real-valued functions on the same domain can be combined pointwise by adding or multiplying their output values.

What it means

The composite function g∘f applies f first (the inner function) and then g (the outer function) to the result, valid as long as the image of f lies in the domain of g.

Components

$f$ inner function, f: A → B
$g$ outer function, g: C → D, with f(A) ⊆ C

What it means

Unlike addition or multiplication, the order in which functions are composed generally matters, producing different results.

What it means

The inverse function undoes f: applying f then f⁻¹ (or vice versa) returns the original input, and it exists mapping the range of f back to the domain.

Components

$f^{- 1}$ inverse function of f

What it means

A function can only be inverted if it is injective (one-to-one); otherwise multiple inputs sharing an output make it impossible to uniquely reverse the mapping.

What it means

Because x² is not injective on all of ℝ (it fails for negative x, since g(f(x)) = |x| ≠ x), √x only serves as its inverse once the domain is restricted to non-negative reals.

What it means

The graph of an inverse function is obtained by reflecting the graph of the original function across the line y = x.

What it means

Since x³ is bijective on all of ℝ, it has a genuine inverse everywhere, namely the cube root function.

Bounds & optima of functions40

What it means

A function is bounded above when its set of output values (image) does not grow without limit; there is some real number that no output exceeds.

Components

$f : A \subseteq R \to R$ the function under consideration
$Im (f)$ the image (range) of f

What it means

A function is bounded below when there is a real number that no output falls below.

Components

$Im (f)$ the image of f

What it means

A function is bounded if its whole image fits between some finite lower and upper number.

What it means

Boundedness extends naturally from single-variable functions to multivariable functions: we simply require the image to be bounded, regardless of the domain's dimension.

Components

$f : R^{n} \to R$ a multivariable real-valued function

What it means

The supremum/infimum of a function is just the supremum/infimum of its set of output values.

Components

$sup_{x \in A} f (x)$ least upper bound of the function's values on A
$in f_{x \in A} f (x)$ greatest lower bound of the function's values on A

What it means

Restricting e^x to (-∞, 2] gives a supremum of e² (attained at x=2) and an infimum of 0 (never attained, since e^x > 0 always).

Components

$e^{2}$ supremum, attained at x = 2
$0$ infimum, approached as x \to -\infty but never attained

What it means

As the input grows, the output never decreases.

What it means

As the input grows, the output strictly grows too — no flat sections allowed.

What it means

As the input grows, the output never increases.

What it means

As the input grows, the output strictly decreases.

What it means

The function takes the same single value everywhere on its domain.

Components

$k$ the fixed constant output value

What it means

A function is monotonic if its direction of change never reverses across its whole domain — it's consistently non-decreasing or consistently non-increasing.

What it means

For a function already known to be increasing, being strictly increasing is equivalent to being one-to-one. Note this equivalence requires the increasing hypothesis: there exist injective functions that are not monotone at all.

Components

$f increasing$ hypothesis: x>y \Rightarrow f(x)\ge f(y)

What it means

A function is strictly increasing exactly when it preserves the ordering of real numbers: comparing inputs gives the same comparison as comparing outputs.

What it means

Concrete checks of the order-preserving criterion: doubling preserves order, a decreasing function like 1/x reverses it.

What it means

A global maximizer is an input where the function's value is at least as large as at any other point in the domain.

Components

$max_{x \in A} f (x)$ the maximum of Im(f), i.e. the global maximum value

What it means

A global minimizer is an input where the function attains a value no larger than at any other point. Together, global maximizers and minimizers are called global optima.

What it means

This upward parabola has a single global minimum at its vertex and increases without bound, so no maximum exists.

What it means

Since e^x is strictly increasing and unbounded with infimum 0 never attained, it has neither a maximum nor a minimum.

What it means

Even a bounded function may fail to have a maximum if its image is an open interval — the supremum is never actually attained.

What it means

On a closed bounded interval a continuous function attains its extrema at the endpoints, but on the corresponding open interval those extrema are not attained since the endpoints are excluded.

What it means

A quartic with two symmetric wells attains its minimum value at two distinct points, illustrating that minimizers need not be unique.

What it means

For a constant function all points achieve the same (maximum and minimum) value, so the set of optima can be infinite.

What it means

There is only one maximum value, but many different inputs could achieve it.

What it means

Argmax/argmin is notation for the whole set of maximizers/minimizers, since more than one point may achieve the optimal value.

Components

$C$ the set over which we optimize

What it means

A strong optimizer beats every other point strictly — no ties allowed — guaranteeing it is the unique optimizer.

What it means

Being the only point achieving the maximum value is equivalent to strictly beating every other point.

What it means

A local optimizer only needs to beat its nearby competitors within some small neighborhood, not the entire domain. Strong local optima are defined analogously with strict inequalities.

Components

$B_{ϵ} (x)$ open ball (interval) of radius epsilon around x

What it means

This function dips to its minimum value at two points and has a local bump (local max, not global) at the origin between them.

What it means

For a strictly increasing function on a closed interval, the endpoints serve as both local and global optimizers.

What it means

A single altered point can create a local maximum out of a discontinuity, while the underlying parabola's infimum of -4 is approached but never reached.

What it means

A multivariable function is bounded exactly when its set of output values is bounded, just as with single-variable functions.

Components

$Im (f)$ image (range) of f

What it means

Examining monotonic behavior in a composite quantity like x²+y² lets us find max/min and decide boundedness quickly.

What it means

A global maximizer/minimizer attains the largest/smallest value of f over the entire domain A ⊆ Rⁿ.

Components

$f (x)$ global maximum/minimum value

What it means

arg max/arg min collects all points in C that achieve the maximum/minimum value of f, not just a single point.

Components

$C \subseteq A$ subset of the domain over which we optimize

What it means

A strong (strict) maximizer/minimizer is the unique point achieving the optimal value; no other point ties it.

What it means

These examples show how to find maxima/minima by analyzing monotonic sub-expressions or using known ranges of trig functions.

What it means

A local maximizer only needs to beat nearby points within some small ball, not the whole domain; local minimizers are defined analogously.

Components

$B_{ϵ} (x)$ open ball of radius ε around x

What it means

Supremum and infimum of a multivariable function are just the sup/inf of its image, exactly as in the one-variable case.

What it means

As x²+y² → ∞ the fraction vanishes so f approaches 1 but never reaches it, giving a supremum with no maximum.

Concave & convex functions39

What it means

Sometimes a function is hard to optimize directly, but composing it with a strictly monotone function (like ln or exp) can turn it into an equivalent, easier problem with the same maximizer.

What it means

Composing with the strictly increasing exponential does not move the location of the maximizer.

What it means

Composing with a strictly increasing function keeps maximizers as maximizers; composing with a strictly decreasing function swaps maximizers and minimizers.

What it means

If x maximizes f, it also maximizes g∘f whenever g is strictly increasing (and the analogous statement holds for minimizers). Note f(x) need not equal g(f(x)) — only the location of the optimum is preserved, and this holds both locally and globally.

Components

$g$ strictly increasing function applied to the outputs of f
$arg max$ set of maximizers

What it means

A firm choosing capital K and labor L to maximize Cobb-Douglas output subject to a budget can apply the strictly increasing ln function to turn a hard product-maximization problem into an easier sum-maximization problem with the same optimizers.

Components

$K, L$ capital and labor inputs
$α$ output elasticity parameter, 0<\alpha<1
$p_{K}, p_{L}$ prices of capital and labor
$λ$ Lagrange multiplier on the budget constraint

What it means

The same monotone-transformation technique used for firms applies to modeling consumer choice, e.g. maximizing a utility function over wine (w) and cheese (c) subject to a budget constraint.

Components

$w, c$ quantities of wine and cheese consumed

What it means

A strictly convex function curves upward everywhere: any chord between two points on the curve stays above the curve itself (except at the endpoints).

What it means

A convex function's chords never dip below the curve, allowing the chord to touch the graph (e.g. on flat/linear portions).

What it means

A strictly concave function curves downward everywhere: any chord between two points stays strictly below the curve.

What it means

A concave function's chords never rise above the curve.

What it means

An inflection point is where the curvature of the function switches from convex to concave (or vice versa) as you pass through it.

Components

$B_{ϵ}^{+} (x)$ right-neighborhood of x of radius epsilon
$B_{δ}^{-} (x)$ left-neighborhood of x of radius delta

What it means

The value of the function at any weighted average (convex combination) of two points is no greater than the same weighted average of the function's values there — the function lies on or below its chords.

Components

$α \in [0, 1]$ weight defining the point on the segment between x and y
$α f (x) + (1 - α) f (y)$ value of the chord (line segment) at that point

What it means

The function's value at a weighted average of two points is at least the weighted average of the function's values — the function lies on or above its chords.

What it means

Both f (a piecewise-linear 'valley') and g (a parabola) are examples used to illustrate convexity: f is convex but not strictly convex (it has flat pieces), while g is strictly convex.

What it means

Strict convexity strengthens the convex inequality to be strict for interior weights, ruling out flat segments in the graph.

What it means

Strict concavity strengthens the concave inequality to be strict for interior weights.

What it means

A set is convex if the line segment joining any two of its points stays entirely inside the set.

What it means

The epigraph is the region on or above the graph of f. A function is convex exactly when this region is a convex set.

Components

$epi (f)$ the set of points lying on or above the graph of f

What it means

The region above the upward parabola is a convex set, confirming that x²+1 is a convex function.

What it means

Since sin(x) oscillates, the area above its graph is not a convex set, so sin(x) is not a convex function on this domain.

What it means

The hypograph is the region on or below the graph of f. A function is concave exactly when this region is convex.

Components

$hypo (f)$ the set of points lying on or below the graph of f

What it means

The downward parabola's hypograph is convex, confirming concavity; more generally, flipping the sign of a function swaps convexity and concavity.

What it means

A straight line satisfies both the convex and concave chord inequalities with equality, so it belongs to both classes.

What it means

Affine functions are defined precisely as those that are simultaneously convex and concave.

What it means

The functions that are both convex and concave are exactly the straight lines: f(x) = mx + b.

Components

$m$ slope of the line
$b$ intercept of the line

What it means

Convexity and concavity are extremely powerful for optimization: they guarantee that any locally optimal point is automatically globally optimal, eliminating the need to search elsewhere. The general version for f: A ⊆ Rⁿ → R is proved later.

What it means

A function of several variables is (additively) separable if it can be written as a sum of functions, each depending on only one of the variables.

Components

$g_{i}$ single-variable function acting only on coordinate x_i
$x = (x_{1}, \dots, x_{n})$ point in the domain A

What it means

Cross terms mixing different variables (like xy) prevent additive separability, since they cannot be split into single-variable pieces.

What it means

On a convex domain A, f is convex if chords lie above the graph, and concave if chords lie below the graph.

Components

$A$ convex subset of Rⁿ (domain of f)
$α x + (1 - α) y$ convex combination of x and y

What it means

The Euclidean norm is convex because the triangle inequality directly gives the defining inequality of convexity.

What it means

This product function fails both the convexity and concavity inequalities, illustrating that many functions are neither.

What it means

A positive-definite quadratic form (plus a constant) is a convex, even strictly convex, function.

What it means

Strict versions require the chord to lie strictly above (convex) or strictly below (concave) the graph for distinct points.

What it means

Affine (linear plus constant) functions are convex and concave but never strictly so, since equality holds along the chord.

What it means

Affine functions (non-vertical lines in R, non-vertical planes/hyperplanes in Rⁿ) are exactly the functions that are simultaneously convex and concave.

What it means

For convex/concave functions on a convex domain, any local optimum is automatically a global optimum, which greatly simplifies optimization.

What it means

Convexity and concavity are preserved under taking non-negative weighted sums of functions, a key tool for building new convex/concave functions.

Components

$ω_{i} \geq 0$ non-negative weights

What it means

Splitting a function into a sum of known convex pieces lets us conclude convexity of the whole using the preservation proposition.

What it means

Unlike non-negative sums, differences, products, and quotients of convex functions need not be convex, as these counterexamples show.

Level sets & quasi-concavity33

What it means

Level curves act like a topographic map, letting us understand a 3D surface through a series of 2D slices at fixed heights.

What it means

The level curve of height k (also called a contour line) is the set of points in the plane where f takes exactly the value k.

Components

$k$ fixed height/level value

What it means

The level-4 curve of this hemisphere function is a circle of radius 3 centered at the origin.

What it means

The level set of level k generalizes level curves to any dimension: it is the set of all domain points mapping exactly to k.

Components

$f^{- 1} (k)$ preimage of the value k, i.e. the level set

What it means

Solving f(x)=k directly gives the level set; in one variable it may be a finite set of points, in two variables a curve.

What it means

Upper contour sets collect points where f is at least k; lower contour sets collect points where f is at most k; their intersection is the level set.

What it means

These examples show how upper/lower contour sets flip in shape depending on whether the function is increasing or decreasing in the relevant quantity.

What it means

Concave functions always have convex upper contour sets, and convex functions always have convex lower contour sets.

What it means

This formalizes that concavity/convexity of a function guarantees convexity of its upper/lower contour sets, respectively.

What it means

The concavity inequality directly shows that any convex combination of two points in the upper contour set stays in the upper contour set.

What it means

Having convex upper and lower contour sets does not imply convexity or concavity of the function; x³ is a counterexample.

What it means

Any strictly increasing or decreasing function on an interval automatically has convex (interval-shaped) contour sets, regardless of curvature.

What it means

A utility function quantifies the usefulness or satisfaction a bundle of goods gives to a consumer; its upper contour sets (u ≥ k) represent bundles giving at least utility k.

Components

$u$ utility function
$(u \geq k)$ bundles delivering at least utility level k

What it means

Mixing (diversifying) two bundles that each guarantee at least utility k should give a mixed bundle with utility at least k too — exactly the statement that upper contour sets of u are convex.

What it means

A function on a convex domain A is quasi-concave if its value at any convex combination of two points is at least the smaller of the two function values.

What it means

A function on a convex domain A is quasi-convex if its value at any convex combination of two points is at most the larger of the two function values.

What it means

A quasi-affine function satisfies both the quasi-concavity and quasi-convexity inequalities simultaneously.

What it means

Strict versions require strict inequality for all distinct points and interior mixing weights, ruling out flat segments at the extreme value.

What it means

This function bottoms out at 0 and rises on both sides, giving it the quasi-convex 'valley' shape; it is not concave/convex in the classical sense despite being classified convex via a special case.

What it means

Its repeated oscillating humps fail every one of these four properties.

What it means

Every concave function is automatically quasi-concave, and every convex function is automatically quasi-convex (with strict versions matching up too); the converse need not hold.

What it means

Quasi-concavity/convexity is strictly weaker than concavity/convexity: x³ satisfies all the quasi properties without being convex or concave.

What it means

Any monotonic function, regardless of curvature, is automatically both quasi-concave and quasi-convex (hence quasi-affine); e.g. eˣ, being strictly increasing, is strictly quasi-concave and strictly quasi-convex.

What it means

Since eˣ is strictly monotone, the monotonicity theorem immediately gives both quasi properties.

What it means

This if-and-only-if result exactly characterizes which functions have convex upper/lower contour sets: precisely the quasi-concave/quasi-convex ones.

What it means

Checking convexity of each contour set directly determines quasi-convexity/quasi-concavity without checking the defining inequality.

What it means

Because |sin(x)| oscillates repeatedly, both its upper and lower contour sets consist of disconnected pieces, failing convexity.

What it means

This bump-shaped function peaks at 0 and decays symmetrically, giving convex upper contour sets (quasi-concave) but non-convex lower contour sets.

What it means

This table summarizes worked classification examples: piecewise linear/absolute-value/odd-power functions illustrate all combinations of the four properties.

What it means

Even in two variables, taking a monotone transformation of a convex expression (like a 5th root) can destroy convexity while preserving quasi-convexity.

What it means

A consumer chooses a bundle (x,y) to maximize a quasi-concave utility function subject to a linear budget constraint, where pₓ, p_y are prices and w is wealth.

Components

$p_{x}, p_{y}$ prices of the two goods
$w$ consumer's wealth/budget

What it means

Strict quasi-concavity of utility, combined with the convex upper contour sets it implies, guarantees the consumer's optimal choice is unique, illustrated graphically where the budget line is tangent to a single upper contour set.

What it means

Composing a quasi-concave function with any strictly increasing function preserves quasi-concavity, mirroring how such compositions preserve maximizers.

Components

$g \circ f$ composition of increasing transformation g with f

Sequences & their limits40

What it means

Informally, a sequence is an ordered, infinite list of real numbers called terms, indexed by the natural numbers.

Components

$x_{n}$ the n-th term of the sequence

What it means

Formally, a sequence is a function from the natural numbers to the reals; the range of this function is the list of terms, and xₙ is called the general term.

Components

$f : N \to R$ sequence viewed as a function
$Im (f)$ range: the set of terms

What it means

The harmonic sequence is the classic example of a sequence whose terms shrink toward zero as n grows.

What it means

Explicit formulas for general terms generate very different qualitative behaviors: converging, growing, diverging to −∞, and oscillating.

What it means

Instead of function notation y = f(n), sequences are usually written with a subscript, such as xₙ, aₙ, etc.

What it means

A recurrence relation defines a sequence via one or more starting values plus a rule expressing each term using earlier terms, rather than an explicit formula in n.

What it means

The same sequence can be described explicitly by a formula or recursively; care is needed with indexing since shifting the index can change the meaning of the recursion.

What it means

The Fibonacci sequence is defined recursively by adding the two previous terms; remarkably, it also has an explicit closed form (Binet's formula) involving the golden ratio.

What it means

As n → ∞, a sequence can be regular (converging to a finite limit or diverging to ±∞) or irregular (e.g., oscillating without settling or blowing up consistently).

What it means

A sequence converges to L if its terms eventually get and stay arbitrarily close to L: for every tolerance ε, all terms beyond some index nε lie within ε of L. Note that nε depends on ε.

Components

$L$ the limit of the sequence
$n_{ϵ}$ threshold index depending on ε, beyond which terms are within ε of L
$B_{ϵ} (L)$ epsilon-neighborhood (open ball) of L

What it means

Two equivalent ways of writing that a sequence converges to L.

What it means

This proof shows formally, using the ε–N definition, that 1 + 1/n converges to 1 by solving the inequality |aₙ − L| < ε for n.

What it means

Since |(−1)ⁿ/n| = 1/n, the same technique as before shows this alternating sequence converges to 0.

What it means

A sequence diverges to +∞ if its terms eventually exceed any given bound k, no matter how large.

Components

$B_{k} (\infty)$ neighborhood of infinity: the interval (k, ∞)

What it means

Since ln(n) grows without bound as n increases, for any bound k we can find an index beyond which ln(n) exceeds k, showing divergence to infinity.

What it means

A sequence diverges to −∞ if its terms eventually fall below any given negative bound k.

Components

$B_{k} (- \infty)$ neighborhood of negative infinity: the interval (−∞, k)

What it means

As n grows, −n³ decreases without bound, so the sequence diverges to −∞; concretely for k = −52, choosing index 5 already suffices.

What it means

A sequence that fails to converge and fails to diverge to ±∞ is called irregular; irregular sequences are often oscillating, such as (−1)ⁿ or (−1)ⁿn².

What it means

A sequence converges to L from below if it approaches L while its terms stay at or below L, e.g. n/(n+1) → 1⁻.

Components

$L^{-}$ notation indicating convergence to L from below

What it means

A sequence converges to L from above if it approaches L while its terms stay at or above L, e.g. n/(n−1) → 1⁺.

Components

$L^{+}$ notation indicating convergence to L from above

What it means

Convergence from a particular direction is a stronger condition than plain convergence: it implies convergence, but a sequence can converge while oscillating around the limit from both sides.

What it means

A sequence is bounded above/below if all its terms stay under/over some fixed real number; it is bounded if both hold.

What it means

A sequence is monotonic if it consistently moves in one direction (non-decreasing or non-increasing); it is strictly monotonic if the inequality is strict, and constant if all terms are equal.

What it means

Since limits only depend on tail behavior, we often only require a property (positivity, monotonicity, a given limit, etc.) to hold from some point onward, not for every single term.

What it means

Even though the sequence increases initially (n² for small n), from n = 11 onward it becomes 2⁻ⁿ, which is decreasing and tends to 0, so both properties hold eventually.

What it means

Although a₀,...,a₄ are ≤ 0, from n = 5 onward the terms are always positive, so the sequence is eventually positive.

What it means

This sequence never settles into any eventual nice behavior: it alternates between blowing up (on even n) and shrinking to zero (on odd n), so it is neither eventually monotonic, bounded, nor convergent.

What it means

Convergence of aₙ to L is equivalent to the distance d(aₙ, L) converging to 0; since distances are never negative, this is the same as converging to 0 from above.

Components

$d (a_{n}, L)$ distance between the term aₙ and the limit L, i.e. |aₙ − L|

What it means

For an eventually positive sequence, diverging to infinity is exactly equivalent to its reciprocal converging to 0 from above. This underlies a useful 'change of variables' trick for computing limits.

Components

$x_{n} > 0 eventually$ the terms are positive from some index onward

What it means

The proof swaps the roles of the bound k and tolerance ε using k = 1/ε (and vice versa), exploiting positivity of xₙ to flip the inequality xₙ > k into 1/xₙ < ε and back.

What it means

If a sequence converges, its limit is unique — a sequence cannot converge to two different finite values at once.

What it means

Assuming two distinct limits leads to a contradiction: choosing ε as half the distance between them forces the triangle inequality to give a value strictly less than itself, which is impossible.

What it means

If a sequence converges to a nonzero limit L, then eventually all terms of the sequence share the same sign as L.

Components

$n^{+}$ threshold index beyond which xₙ has the same sign as L

What it means

Taking ε to be half the (nonzero) limit L forces the terms, once close enough to L, to lie strictly on the same side of 0 as L, so their product with L is positive.

What it means

Every convergent sequence must be bounded; convergence is a stronger property than boundedness.

What it means

A convergent sequence is eventually trapped in a neighborhood of its limit (so the infinite tail is bounded), and the only remaining terms are a finite set, which is automatically bounded; combining the two bounds bounds the whole sequence.

What it means

Boundedness alone does not guarantee convergence; the converse of the previous theorem fails because of irregular (oscillating) sequences. However, bounded regular sequences do converge.

What it means

Every monotonic sequence is regular: it either converges (if bounded) or diverges to ±∞ (if unbounded); it can never be irregular/oscillating.

What it means

The key idea is completeness of the reals: a bounded monotonic sequence converges to its supremum (or infimum), because eventually the sequence must get arbitrarily close to this least upper bound; an unbounded monotonic sequence diverges since it must eventually exceed every bound.

Components

$sup {x_{n}}$ least upper bound of the sequence, which exists by completeness of R when bounded above

What it means

Regularity (here, convergence) does not imply monotonicity: this alternating sequence converges to 0 while continually switching between increasing and decreasing.

Computing limits of sequences59

What it means

Power sequences nᵅ are monotonic (hence regular). They diverge to infinity for positive exponent, are constantly 1 for exponent zero, and converge to 0 for negative exponent.

Components

$n^{α}$ power sequence with exponent α

What it means

The exponential sequence qⁿ behaves differently depending on the base q: it diverges for q>1, is constant for q=1, converges to 0 for |q|<1, and oscillates without a limit (irregular) whenever q≤-1.

Components

$q^{n}$ exponential sequence with ratio q

What it means

Logarithmic sequences (ln n)ᵅ mirror the behavior of power sequences: they diverge for positive exponent, equal 1 for exponent zero, and converge to zero for negative exponent.

Components

$(ln n)^{α}$ logarithmic sequence with exponent α

What it means

A geometric sequence multiplies the previous term by a fixed ratio q at each step, starting from an initial value α; e.g. α=3, q=2 gives xₙ=3·2ⁿ.

Components

$α$ initial value
$q$ common ratio

What it means

An arithmetic sequence adds a fixed common difference q to the previous term at each step, starting from an initial value α; e.g. α=3, q=5 gives xₙ=3+5n.

Components

$α$ initial value
$q$ common difference

What it means

If two sequences are regular (converge to a real number or ±∞), then their sum, product, and quotient are also regular and equal the corresponding operation on the limits, unless the combination produces an indeterminate form, in which case more analysis is needed.

Components

$R$ extended real line, \mathbb{R} \cup \{\infty,-\infty\}
$L, H$ limits of the two sequences

What it means

Since n diverges to infinity and 1/n converges to 0, their sum is not an indeterminate form and diverges to infinity by the sum rule.

What it means

Both terms diverge to +∞, and since +∞+∞ is not indeterminate, the sum also diverges to +∞.

What it means

Both eⁿ and n diverge to infinity, so their difference is an indeterminate form and cannot be resolved by the basic operations theorem alone; further tools (like the hierarchy of infinities in Lecture 14) are needed.

What it means

The product of two sequences both diverging to +∞ diverges to +∞, since this is not an indeterminate form.

What it means

One factor diverges while the other tends to 0, producing an indeterminate product that requires additional analysis to resolve.

What it means

The product of two sequences both converging to 0 converges to 0, since 0·0 is not an indeterminate form.

What it means

Even though each ratio starts as the indeterminate form ∞/∞, canceling common factors of n reveals the true limit, which can be ∞, 0, or a finite nonzero number depending on the relative degrees.

What it means

The proof uses the standard ε-N definition of convergence: to control the sum's distance from L+H, we control each sequence's distance from its own limit within ε/2 and use the triangle inequality.

Components

$ϵ$ arbitrary tolerance
$n_{ϵ}$ index beyond which the approximation holds

What it means

If a sequence is trapped between two other sequences that both converge to the same limit L, then it too must converge to L — useful for irregular sequences like sin(n) that cannot be handled by the operations theorem.

Components

$y_{n} \leq x_{n} \leq z_{n}$ squeeze inequality, holding eventually (for large n)

What it means

Once both bounding sequences are within ε of L, the trapped sequence xₙ is forced to also be within ε of L, so it converges to L.

What it means

Since sin(n) is bounded between -1 and 1 while n grows without bound, dividing by n squeezes sin(n)/n between two sequences both converging to 0, forcing sin(n)/n to converge to 0 even though sin(n) itself is irregular.

What it means

If the ratio of consecutive terms (in absolute value) converges to a number strictly less than 1, then the sequence must converge to 0 — a powerful tool for sequences involving factorials or exponentials.

Components

$\frac{x _{n + 1}}{x _{n}}$ ratio of consecutive terms' absolute values
$L < 1$ limit of the ratio, strictly less than 1

What it means

Since the ratio of successive terms approaches 1/3, which is less than 1, the ratio criterion guarantees the sequence n³/3ⁿ converges to 0, even though both numerator and denominator individually diverge.

What it means

Formal names for the two extreme types of behavior we compare: sequences diverging to ±∞ (infinities) and sequences converging to 0 (infinitesimals).

What it means

aₙ is 'slower' to reach infinity than bₙ; we say aₙ is negligible with respect to bₙ.

What it means

We write aₙ = O(bₙ) (read 'aₙ is little-o of bₙ') when aₙ is negligible with respect to bₙ, i.e. their ratio tends to 0.

Components

$O (b_{n})$ little-o notation: negligible with respect to b_n

What it means

aₙ is 'faster' to reach infinity than bₙ; equivalently bₙ is negligible with respect to aₙ, so bₙ = O(aₙ).

What it means

When the ratio of two infinities tends to a nonzero finite constant, they grow at comparable ('same order') speeds, denoted aₙ ≍ bₙ.

Components

$a_{n} ≍ b_{n}$ a_n and b_n are of the same order (comparable)

What it means

Two sequences are asymptotically equivalent when their ratio tends to exactly 1, meaning they behave identically in the limit; equivalence of an ~ bn transfers convergence and limit value between the two sequences.

Components

$a_{n} \sim b_{n}$ a_n asymptotically equivalent to b_n

What it means

Sometimes the ratio of two infinities reduces to an irregular (oscillating) sequence with no limit, in which case the two infinities simply cannot be ranked against each other.

What it means

Among power sequences, the one with the larger exponent is always the infinity of higher order.

What it means

Among exponential sequences, the one with the larger base is the infinity of higher order.

What it means

Among logarithmic power sequences, the larger exponent gives the infinity of higher order, exactly analogous to power sequences.

What it means

Any exponential sequence with base greater than 1 grows faster than any power sequence, no matter how large the exponent; this is confirmed using the ratio criterion together with the fact that a sequence diverges to infinity exactly when its reciprocal converges to 0.

What it means

Any power sequence with a positive exponent grows faster than any power of the logarithm, regardless of the exponents involved.

What it means

The preceding comparisons establish a general ranking: exponential growth beats polynomial growth, which in turn beats logarithmic growth.

What it means

Factorial growth outpaces any exponential growth, and nⁿ in turn outpaces factorial growth, extending the hierarchy further.

What it means

Even the smallest positive power of ln(n) beats ln(ln n), so ln(ln n) sits at the very bottom of the hierarchy of infinities.

What it means

A complete ranking of common sequences by growth rate as n → ∞, from slowest (double logarithm) to fastest (n to the n).

What it means

Little-o terms combine algebraically much like error terms: sums and scalar multiples of negligible terms remain negligible at the same order, and the dominant term absorbs a strictly smaller-order term.

What it means

Asymptotic equivalence is preserved under multiplication of equivalent sequences and under raising to a positive power, which makes it a very convenient tool for simplifying limits.

What it means

Two sequences are asymptotically equivalent exactly when they differ from each other only by a term that is negligible relative to them — i.e. they agree up to lower-order corrections.

What it means

Since a sequence plus a negligible correction is asymptotically equivalent to the sequence itself, products and quotients of such sums are asymptotically equivalent to the products/quotients of the leading terms — the key technique used to resolve indeterminate forms.

What it means

By identifying the dominant (highest-order) term in numerator and denominator and treating all lower-order terms as negligible, the limit reduces to the ratio of leading coefficients: 3/5.

What it means

Since ln n is negligible compared to nⁿ, the difference is dominated entirely by −nⁿ, which diverges to −∞.

What it means

The numerator's dominant term n² is itself negligible compared to eⁿ in the denominator, so the whole expression tends to 0.

What it means

Both numerator and denominator are dominated by their cubic terms, so the limit is the ratio of the leading coefficients, 2.

What it means

Bounded oscillating terms like (−1)ⁿln(n)³ and sin(n) are negligible compared to n² in the numerator, and n², n³ are negligible compared to eⁿ in the denominator, so the ratio reduces to n²/eⁿ → 0.

What it means

Just as with infinities, we can compare the relative 'speed' of two sequences both tending to zero by examining the limit of their ratio; the case of 0⁻ is entirely analogous.

What it means

aₙ is 'faster' to reach 0 than bₙ, i.e. aₙ is negligible relative to bₙ.

What it means

aₙ is 'slower' to reach 0 than bₙ, meaning bₙ is negligible relative to aₙ.

What it means

When the ratio of two infinitesimals tends to a nonzero constant, they approach zero at comparable rates, written aₙ ≍ bₙ.

What it means

Just as with infinities, two infinitesimals are asymptotically equivalent when their ratio tends to exactly 1.

What it means

As with infinities, the ratio of two infinitesimals can reduce to an irregular sequence, meaning the two infinitesimals cannot be ranked.

What it means

The hierarchy of infinities and the hierarchy of infinitesimals are mirror images of each other: aₙ being a higher-order infinity than bₙ is equivalent to 1/aₙ being a higher-order infinitesimal than 1/bₙ.

What it means

The hierarchy of infinitesimals is exactly the reverse of the hierarchy of infinities: reciprocals of the fastest-growing sequences are the fastest to reach zero.

What it means

The dominant terms 3ⁿ and 4ⁿ control numerator and denominator, giving a ratio of (3/4)ⁿ which converges to 0 since 3/4 < 1.

What it means

Rewriting n^(1/n) using the exponential/logarithm identity converts the problem to computing lim(ln n)/n, which is 0 because n grows faster than ln n; exponentiating back gives limit e⁰=1.

What it means

The sequence (1+1/n)ⁿ is monotonically increasing and bounded above, so by the monotone convergence theorem it converges to its supremum.

What it means

The sequence (1+1/n)ⁿ converges to Euler's number e, and more generally (1+a/n)ⁿ converges to e^a for any real constant a — a fundamental limit used throughout calculus and compound interest problems.

Components

$e$ Euler's/Napier's number, approximately 2.71828

What it means

A vector-valued sequence assigns to each natural number n a vector in ℝᵐ, generalizing the notion of a real-number sequence.

Components

$x_{n} \in R^{m}$ the n-th vector term, with m components

What it means

A vector sequence converges to a limit vector L when, eventually, the distance between xₙ and L becomes arbitrarily small; conveniently, this holds if and only if every coordinate sequence converges to the corresponding coordinate of L.

Components

$∥ x_{n} - L ∥$ Euclidean distance between x_n and L
$B_{ϵ} (L)$ ε-neighborhood (open ball) around L

What it means

To find the limit of a vector sequence we take the limit of each coordinate separately: the first coordinate uses the hierarchy (ln n negligible relative to 2n, so the exponent goes to −∞), the second uses the generalized e^a limit, and the third is a simple rational limit.

Series62

What it means

Naively summing an infinite sequence of ±1 terms by different groupings gives contradictory answers (0, 1, 1/2), showing that infinite sums need a rigorous definition before being manipulated freely.

Components

$a_{n}$ general term of the sequence
$S$ purported sum of the series

What it means

Walking half the remaining distance repeatedly corresponds to summing an infinite geometric sequence of distances; even though infinitely many strictly positive terms are added, the total is the finite distance of 1 meter.

Components

$(\frac{1}{2})^{n}$ distance walked at step n

What it means

Given a sequence aₙ, the series with general term aₙ is the formal infinite sum of its terms.

Components

$a_{n}$ general term of the sequence
$\sum_{n = 0}^{\infty} a_{n}$ the series generated by aₙ

What it means

From a sequence aₙ we build a new sequence sₙ, the partial sums, where each new term adds the next aₙ to the previous partial sum. A series is precisely this sequence of partial sums.

Components

$s_{n}$ n-th partial sum
$a_{n}$ n-th term added

What it means

If the limit of the sequence of partial sums exists, the series is said to have that value; convergence of a series reduces to convergence of the sequence of partial sums.

Components

$s_{n}$ sequence of partial sums

What it means

This resolves the opening paradox: since the partial sums oscillate between 1 and 0 without converging, the series simply has no value — none of 0, 1, or 1/2 is correct.

What it means

Since a series is just a sequence of partial sums, its behavior (convergent, divergent to ±∞, or irregular) is classified exactly as sequence behavior is classified, applied to sₙ.

Components

$s_{n}$ sequence of partial sums

What it means

Given a geometric sequence aₙ = aqⁿ, the geometric series is the infinite sum of its terms, with a the first term and q the common ratio.

Components

$a$ first term
$q$ common ratio

What it means

When the common ratio exceeds 1, the terms themselves grow without bound, so the partial sums also diverge.

What it means

When q = 1 every term equals a, so the partial sum after n terms is simply na, which diverges as n grows (unless a = 0).

What it means

For q more negative than −1, the magnitude of the terms grows without bound while alternating sign, making both the sequence and its partial sums irregular and unbounded.

What it means

At q = −1 the terms have constant magnitude but alternate sign forever, so the sequence and partial sums are bounded but never settle to a limit.

What it means

When the common ratio is strictly between −1 and 1, the terms shrink to zero fast enough that the amount added at each step vanishes and the partial sums converge to a finite limit.

What it means

This fully classifies the geometric series (with a = 1): it diverges to infinity for q ≥ 1, converges to 1/(1−q) for |q| < 1, and is irregular for q ≤ −1.

Components

$q$ common ratio

What it means

The key algebraic trick is writing (1−q)sₙ as a telescoping difference to get the closed form sₙ = (1−qⁿ⁺¹)/(1−q); the limit behavior of qⁿ⁺¹ in each range of q then determines the behavior of the series.

Components

$s_{n} = \frac{1 - q ^{n + 1}}{1 - q}$ closed form of the n-th partial sum for q ≠ 1

What it means

If the sum starts at n = n₀ instead of 0, the value is rescaled by q raised to the starting index, obtained by factoring out qⁿ⁰ and applying the standard formula to the remaining series starting at 0.

Components

$n_{0}$ starting index of summation

What it means

Factoring out the first term's power converts a series starting at n=1 into a series starting at n=0, which can then be evaluated with the standard geometric series formula.

What it means

The fully general geometric series formula: for any first coefficient a and any starting index n₀, the sum equals the first term of the series divided by one minus the common ratio.

Components

$a$ scaling coefficient
$a q^{n_{0}}$ first term actually appearing in the sum

What it means

Applying the geometric series formula confirms rigorously that the total distance walked in Zeno's paradox is exactly 1 meter, resolving the paradox.

What it means

A worked example applying the most general geometric series formula with a = 2, q = 1/3, n₀ = 0.

What it means

Since series are limits of partial sums, they inherit the linearity properties of limits: convergent series can be added term-by-term and scaled by constants.

Components

$c$ scalar constant

What it means

The harmonic series is the sum of reciprocals of the positive integers, a special series whose terms tend to zero yet whose sum diverges.

What it means

Despite its terms shrinking to zero, the harmonic series is positively divergent — the terms do not shrink fast enough for the sum to be finite.

What it means

By grouping consecutive terms into blocks of doubling length (1 term, then 2, then 4, then 8, ...), each block sums to strictly more than 1/2; since infinitely many such blocks exist, the partial sums grow without bound.

What it means

This is a crucial cautionary example: the terms of a series tending to zero is not enough to guarantee convergence of the series — how fast they tend to zero matters.

What it means

The Mengoli series is a classic telescoping series whose partial sums collapse via partial fractions [1/n − 1/(n+1)], yielding a sum of exactly 1; left as a homework exercise.

What it means

This recalls the harmonic vs. geometric series comparison, motivating the search for necessary and sufficient conditions for convergence beyond simply aₙ → 0.

What it means

Convergence of a series forces its terms to tend to zero; however, this condition is necessary but not sufficient (the harmonic series is the standard counterexample to sufficiency).

What it means

The logically equivalent contrapositive gives a practical divergence test: if the terms do not tend to zero, the series cannot converge.

What it means

Using asymptotic equivalence to compute the limit of the general term shows it is nonzero, so by the contrapositive of the necessary condition the series cannot converge (though this alone does not show it diverges to infinity rather than being irregular).

What it means

A series that is eventually non-negative can never be irregular: it must either converge to a finite value or diverge to +∞, since its partial sums are eventually non-decreasing.

Components

$regular$ either convergent or divergent to +infinity

What it means

Combining non-negativity (hence regularity, so it must converge or diverge) with the failure of the necessary condition (so it cannot converge) forces the conclusion that the series diverges to infinity.

What it means

Since the terms are eventually non-negative (regular) but do not tend to zero, the series must diverge to infinity.

What it means

This generalizes the harmonic series (α = 1) by raising n to an arbitrary real power α in the denominator.

Components

$α$ exponent parameter

What it means

The generalized harmonic (p-)series converges exactly when the exponent exceeds 1; it does not tell us the value of the sum, only whether one exists.

What it means

This further generalizes the generalized harmonic series by adding a logarithmic factor, reducing to it when β = 0.

Components

$α, β$ exponent parameters

What it means

This series converges when α exceeds 1 regardless of β, or when α equals 1 exactly if β exceeds 1; it diverges when α is below 1, or when α equals 1 with β at most 1.

What it means

Two eventually positive series with asymptotically equivalent general terms always share the same convergence/divergence behavior, though not necessarily the same sum.

Components

$a_{n} \sim b_{n}$ asymptotic equivalence, i.e. \lim a_n/b_n = 1

What it means

Since the general term is asymptotically equivalent to a constant multiple of a convergent generalized harmonic series (α = 2), the asymptotic comparison criterion gives convergence.

What it means

Even with a bounded oscillating term like cos n in the denominator, it is dominated (Big-O) by the leading terms, so the asymptotic comparison still yields convergence.

What it means

The general term is asymptotically equivalent to 1/n, so by comparison to the divergent harmonic series, the given series diverges to infinity.

What it means

When exponential terms dominate polynomial ones, the series behaves like a convergent geometric series since 3/π < 1.

What it means

If one non-negative series is termwise bounded above by another, convergence of the larger series forces convergence of the smaller, and divergence of the smaller forces divergence of the larger; this does not determine the value of either sum.

What it means

Bounding 1/e^{n²} above by the convergent series 1/n² (a comparison with different starting index n=0 vs n=1 does not affect convergence, since finitely many terms never do) shows the given series converges.

What it means

Since 1/ln(ln n) is bounded below by the divergent harmonic series terms, the comparison criterion forces this series to diverge too, even though its terms tend to zero (just too slowly).

What it means

Comparing each term to the next via their ratio, this test tells us convergence or divergence based on whether the terms eventually shrink geometrically fast; it does not reveal the sum's value.

Components

$\frac{a _{n + 1}}{a _{n}}$ ratio of consecutive terms

What it means

When the limiting ratio equals exactly 1, the ratio test gives no information and another method must be used (e.g. the harmonic vs. generalized harmonic series both have ratio → 1 but differ in behavior).

What it means

The factorial growth in the denominator makes the ratio of consecutive terms tend to 0, well below 1, so the ratio criterion confirms convergence.

What it means

The exponential series sums the reciprocals of factorials; it is a fundamental series whose value turns out to be Euler's number e.

What it means

Applying the ratio criterion to the exponential series shows the ratio of consecutive terms vanishes, confirming convergence.

What it means

The exponential series with 1/n! sums exactly to Euler's number e, and more generally, substituting xⁿ/n! gives the Taylor series for eˣ at every real x.

Components

$e$ Euler's/Napier's number, ≈ 2.71828
$e^{x}$ the exponential function evaluated at x

What it means

All the previous tests (comparison, ratio, asymptotic) assumed eventually non-negative terms; we now extend the theory to series whose terms may have mixed or indefinite sign.

What it means

Factoring out the constant −1 reduces this to the (negatively scaled) harmonic series, which diverges; if a series can be written as −Σbₙ where Σbₙ converges, then the original series converges too.

What it means

Any all-negative series can be handled by factoring out −1 and analyzing the resulting non-negative series with the earlier tests.

What it means

These examples have terms that permanently switch sign in no simple eventual pattern, motivating the need for a genuinely new tool: absolute convergence.

What it means

Given any series with terms of arbitrary sign, we can form the associated series of absolute values, which is always a non-negative series and hence regular, so the earlier tests apply to it directly.

What it means

A series converges absolutely if the series of its absolute values converges; this is a stronger condition than ordinary convergence.

What it means

Absolute convergence is a sufficient condition for ordinary convergence, giving a powerful tool for series with mixed-sign terms: if the absolute-value series converges (checkable with non-negative-series tests), the original series converges too.

What it means

Taking absolute values removes the alternating sign, reducing to the convergent generalized harmonic series with α = 2, so the original series converges absolutely.

What it means

Since |sin n| ≤ 1, the absolute-value series is bounded above by the convergent Σ1/n², so by the comparison criterion it converges, giving absolute convergence and hence convergence of the original series.

What it means

The converse of the absolute convergence theorem is false: some series converge without converging absolutely. Such series are called conditionally convergent.

What it means

The alternating harmonic series converges in the ordinary sense, but its absolute-value series is exactly the divergent harmonic series, so it does not converge absolutely — it is the canonical example of conditional convergence.

Limits of functions58

What it means

Away from x=1 this function equals x+1, so it should 'behave like' x+1 near x=1 even though it is undefined there. Limit points of a set need not belong to the set.

Components

$x = 1$ point excluded from the domain but still a limit point of it

What it means

There are four combinations depending on whether we approach a finite point or infinity, and whether the function value approaches a finite number or infinity. The limit may also fail to exist at all.

What it means

f(x) can be made arbitrarily close to L by taking x sufficiently (but not exactly) close to x₀. This is the classical ε–δ definition, with d(x,y)=|x−y|.

Components

$ϵ$ target closeness of f(x) to L
$δ$ required closeness of x to x_0
$d (x, y) = ∣ x - y ∣$ distance on the real line

What it means

Choosing δ = ε makes the ε–δ definition hold, verifying the limit directly from the definition.

What it means

A unified definition valid for all four cases, using open neighborhoods instead of distances; this also covers neighborhoods of ±∞, e.g. B_ε(∞) = (ε,∞).

Components

$B_{ϵ} (L)$ neighborhood of L (possibly infinite)
$B_{δ} (x_{0})$ neighborhood of x_0 (possibly infinite)

What it means

Given any neighborhood of infinity, we found a matching neighborhood of 2 that guarantees f(x) lands in it, confirming the limit is infinite.

What it means

Points sufficiently close to x₀ (but not equal to it) make f(x) larger than any prescribed M.

Components

$M$ arbitrarily large target bound for f(x)

What it means

For sufficiently large x, f(x) is arbitrarily close to L.

What it means

By reusing the known sequence limit for 1/2ⁿ → 0, we transfer that estimate to the continuous variable x to establish the function limit.

What it means

The limits at +∞ and −∞ can be different finite values for the same function.

What it means

For sufficiently large x, f(x) becomes arbitrarily large as well.

What it means

A simple example of an infinite limit point at infinity in both directions.

What it means

Since the two-sided definition requires all nearby x (both sides) to behave consistently, functions that differ on each side of a point need one-sided limits.

What it means

One-sided limits restrict attention to x approaching x₀ only from the right (x₀⁺) or only from the left (x₀⁻).

What it means

Only points on the right side of x₀ need be controlled to land f(x) inside the target neighborhood. Left-sided definitions are analogous.

What it means

The only difference from the two-sided ε–δ definition is restricting x to the interval (x₀, x₀+δ) rather than a full punctured neighborhood.

What it means

For x just to the right of x₀, f(x) can be forced above any bound M.

What it means

The left and right limits can be entirely different in character (one infinite, one finite).

What it means

Even when both one-sided limits exist, they need not be equal — so the two-sided limit fails to exist here.

What it means

The two-sided limit exists and equals L exactly when both one-sided limits exist and agree on the value L.

What it means

As x→0, 1/x oscillates through all real values infinitely often, so sin(1/x) oscillates between −1 and 1 without settling, so no limit (one- or two-sided) exists.

What it means

Since both rationals and irrationals are dense, every neighborhood of any point contains points where f=1 and points where f=0, so f(x) never settles to a single value.

What it means

f approaches L while staying below (or equal to) L as x grows large; other 'from above/below' variants (at finite points, one-sided) are defined analogously.

What it means

For large enough x, f(x) is within ε of L but never exceeds L, capturing approach 'from below'.

What it means

A vertical line at x=x₀ is an asymptote whenever the function blows up (to +∞ or −∞) approaching x₀ from at least one side; it can be complete (both sides blow up) or incomplete (only one side).

What it means

Only one side (the left) blows up to infinity near x=1, while the right side stays finite, so the asymptote only touches the curve on one side.

What it means

A horizontal line y=L is an asymptote if the function approaches L as x grows without bound in either direction; a function may have two different horizontal asymptotes and may even cross one (e.g. sin(x)/x crosses y=0).

What it means

A worked example locating both a complete vertical asymptote at x=1/2 and a horizontal asymptote at y=3/2 for a rational function.

What it means

This connects function limits to sequence limits: f has limit L at x₀ exactly when every sequence approaching x₀ (never equal to x₀) has images converging to L. This lets us reuse all sequence-limit theorems for functions.

What it means

A function cannot have two different finite limits at the same point; the limit, if it exists, is unique.

What it means

Uniqueness of limits for functions follows immediately from uniqueness of limits for sequences via the sequential characterization theorem.

What it means

If the limit is nonzero, then near x₀ the function value must have the same sign as the limit L.

What it means

The squeeze theorem for functions: if f is trapped between two functions with the same limit, f shares that limit.

What it means

Since sin(x) is bounded between −1 and 1, dividing by x squeezes sin(x)/x between two functions that both tend to 0.

What it means

This is the roadmap for computing limits of functions: recognize elementary function behavior, apply algebra-of-limits rules, and fall back on notable limits or substitutions when facing indeterminate forms.

What it means

Sums, products, and quotients of functions with known limits have limits equal to the corresponding sum, product, or quotient of those limits, provided we don't hit an indeterminate form like ∞−∞ or 0/0.

What it means

The isolated statement for addition, proved directly using the sequential characterization of limits.

What it means

The strategy is to reduce the function-limit statement to sequences (using Theorem (*): limₓ→x₀f(x)=L ⟺ xₙ→x₀ ⟹ f(xₙ)→L), apply the already-proven sequence addition rule, then translate back.

What it means

Even though the algebra-of-limits theorem cannot be applied directly (cos x has no limit), boundedness of cos(x) combined with eˣ→∞ still forces the sum to diverge to infinity.

What it means

Since our sequence tools require n→∞, substitutions like t=1/x (for x→0⁺) or t=−x (for x→−∞) convert any type of limit into one where the new variable tends to infinity.

What it means

Substituting t=1/x converts the limit into one at infinity, which is then evaluated using the sequence order-of-growth fact ln(n)=O(n), approaching 0 from below since ln(1/n)<0.

What it means

Substituting t=−x turns a limit at −∞ into one at ∞, then the known growth-order fact n=O(eⁿ) gives the value 0, approached from below.

What it means

f∼g means f and g behave the same way (to leading order) near x₀; f=O(g) means f is negligible compared to g near x₀. These generalize sequence notation to functions, and x₀ may be finite or ±∞.

Components

$x_{0}$ any limit point, finite or infinite, at which the comparison is made

What it means

Which function is 'bigger' depends entirely on where you are taking the limit: x² dominates x at infinity, but x dominates x² near zero. The hierarchy of infinities/infinitesimals is location-dependent.

What it means

Keeping only the dominant term in numerator and denominator as x→∞ simplifies the ratio to 1/x, whose limit is immediate.

What it means

Near 0, the lowest-order (dominant) terms are 2x in both numerator and denominator, giving the limit 1/2 — the opposite dominant term compared to the x→∞ case.

What it means

A continuous-variable analogue of the sequence definition of e; setting a=1 recovers limₓ→∞(1+1/x)ˣ=e.

What it means

Obtained from the previous notable limit by a change of variable t=1/x, converting x→0 into t→∞.

What it means

Near x=0, eˣ−1 behaves exactly like x to leading order; equivalently eˣ is well-approximated by 1+x.

What it means

Near x=0, ln(1+x) behaves like x to leading order.

What it means

Near x=0, raising (1+x) to a power a produces a change approximated by ax; this generalizes the binomial approximation.

What it means

The fundamental trigonometric limit: near 0, sin(x) is well-approximated by x itself.

What it means

Near 0, cosine deviates from 1 by approximately x²/2, giving the standard second-order approximation for cosine.

What it means

Recognizing √(1+x)−1 as the a=1/2 case of the power notable limit immediately gives the answer 1/2.

What it means

A change of variable transforms this limit at infinity into the fundamental notable limit sin(t)/t → 1 as t→0.

What it means

Substituting t=x² reduces this to the standard notable limit ln(1+t)/t → 1.

What it means

Applying asymptotic equivalence (rather than direct substitution) and simplifying the resulting ratio gives the limit 0.

What it means

Combining two notable limits (for eˣ−1 and ln(1+x)) via asymptotic equivalence reduces a compound indeterminate ratio to a simple limit of x.

Continuity57

What it means

A function is continuous at a point if its limit there equals its actual value there — no gap between what the function approaches and what it equals.

Components

$x_{0}$ point of the domain (must be a limit point of A)
$lim_{x \to x_{0}} f (x)$ value the function approaches near x_0

What it means

At an isolated point there is no way to take a limit (no nearby points to approach from), so continuity there is simply declared true by convention.

What it means

A function is continuous on its whole domain when it is continuous at each and every point of that domain.

What it means

Continuity is a property tied to points where the function is actually defined; it makes no sense to ask about continuity at a point outside the domain.

What it means

Even though 1/x blows up as x approaches 0, since 0 is excluded from the domain, the function is continuous everywhere it is actually defined.

What it means

The two pieces do not meet up at x = 0 (left limit 1, right limit 3), so the function jumps there and fails continuity.

What it means

Checking one-sided limits at the boundary point (here x=1, which is in the domain) reveals a mismatch, so the function is discontinuous there.

What it means

Since 2 and 4 are not adjacent (there is a gap between them), we only need continuity within each piece; there is no point where the two pieces must match up, so the function is continuous.

What it means

Once we force the function to be defined at x=0, we must check continuity there, and since the limit does not exist (it blows up), the function fails to be continuous.

What it means

Discontinuities are classified into three types based on the behavior of one-sided limits: holes can be patched, jumps and essential discontinuities cannot.

What it means

The limit exists and is finite, but it does not equal the actual function value — like a single missing or misplaced point that could be 'fixed' by redefining f(x_0).

What it means

Both one-sided limits are finite but disagree — the graph literally jumps from one height to another, and no single value of f(x_0) can fix this.

What it means

At least one side blows up to infinity or oscillates without settling — the worst kind of discontinuity, impossible to fix by any redefinition.

What it means

The limit as x approaches 0 is 0, but the function was defined to equal 1 there — a single misplaced point, i.e. a removable hole.

What it means

Both one-sided limits blow up to infinity (in opposite directions), so this cannot be patched by redefining the function value — an essential discontinuity.

What it means

As x approaches 0, sin(1/x) oscillates infinitely often between -1 and 1 without settling on any value, so the limit fails to exist — an essential discontinuity.

What it means

Since rationals and irrationals are both dense, every neighborhood of any point contains values of f equal to both 0 and 1, so no limit ever exists, making every point an essential discontinuity.

What it means

To make a piecewise function continuous at the boundary point, set the two one-sided expressions equal at that point and solve for the unknown parameter.

What it means

A function has a limit at a point exactly when every sequence approaching that point (without hitting it) has its images converge to the same value — this bridges the language of function limits and sequence limits.

Components

$x_{n} \to x_{0}, x_{n} \neq = x_{0}$ sequence approaching x_0 but never equal to it

What it means

Continuity at a point is equivalent to: whenever inputs converge to that point (this time sequences may equal x_0), the outputs converge to the function's actual value there.

What it means

Merely requiring that the image sequence converges (without specifying the limit must equal f(x_0)) is a strictly weaker condition and does not imply continuity — the images could converge to the wrong value.

What it means

Familiar building-block functions (polynomials, rational functions, roots, exponentials, logarithms, trigonometric functions, and their combinations) are automatically continuous everywhere they are defined.

What it means

Sums, differences, products, and quotients (with nonzero denominator) of continuous functions remain continuous at the point in question — continuity is preserved under the basic algebraic operations.

What it means

Even though f and g are individually discontinuous at 0 (both jump), their sum is the constant 0, their product is the constant -1, and their quotient is constant -1 — all continuous. This shows combining functions can 'cancel out' discontinuities.

What it means

Composing two continuous functions (where the composition makes sense) yields another continuous function; this justifies computing limits of composite continuous functions by direct substitution.

Components

$g \circ f$ composition, (g\circ f)(x) = g(f(x))

What it means

For a continuous composite function, you can compute the limit simply by plugging in x_0 — no special technique required.

What it means

Since both cosine and the exponential function are continuous, we can evaluate the limit of their composition by direct substitution of x = π.

What it means

A continuous function on a compact (closed and bounded) domain always attains both a maximum and a minimum value somewhere on that domain — existence of optima is guaranteed.

Components

$A compact$ A is closed and bounded
$max_{z \in A} f (z)$ the maximum value attained by f on A

What it means

Each hypothesis in Weierstrass's Theorem is essential; the theorem only tells us that a max/min exists, not what its value is.

What it means

Although [0,1] is compact, f is discontinuous at the endpoints (a hole discontinuity), so the supremum 1 is never actually attained — Weierstrass fails without continuity.

What it means

Even though f is continuous, the open interval (0,1) is not closed, so f gets arbitrarily close to 0 and 1 but never actually attains those values.

What it means

Since the domain R is unbounded, arctan approaches but never reaches its horizontal asymptotes ±π/2, so no actual max or min exists.

What it means

If utility U is continuous and prices p_w, p_c are strictly positive with finite budget m, the budget set is compact, so by Weierstrass a utility-maximizing bundle always exists.

Components

$p_{w}, p_{c}$ strictly positive prices of wine and cheese
$m$ total budget available

What it means

If a continuous function on a closed interval takes values of opposite sign at the two endpoints, it must cross zero somewhere in between; if it's also strictly monotone, that zero is unique.

Components

$f (a) \cdot f (b) \leq 0$ f(a) and f(b) have opposite signs (or one is zero)

What it means

This worked example illustrates the standard three-step check (compact interval, continuity, opposite-sign endpoints) required to invoke Bolzano Theorem and conclude a root exists.

What it means

If two continuous curves swap which one is on top somewhere over an interval, they must cross; applying this with f = demand and g = supply guarantees a market-clearing price exists.

What it means

The market equilibrium theorem is proven by simply applying Bolzano Theorem to the difference function h = f - g, converting 'f equals g' into 'h has a zero'.

What it means

A continuous function on a closed interval hits every value between its minimum and maximum at least once — it cannot 'skip' any intermediate value.

Components

$m, M$ minimum and maximum values of f on [a,b] (exist by Weierstrass)

What it means

Combining Weierstrass (min/max exist) and Darboux (every value in between is hit), the entire image of a continuous function on a closed interval is itself a closed interval from the min to the max.

What it means

For continuous functions defined on an interval, being one-to-one is exactly equivalent to being strictly increasing or strictly decreasing — a much stronger link than for general functions.

What it means

This function is injective (one-to-one) without being monotone, showing the equivalence between injectivity and strict monotonicity breaks down once continuity is dropped.

What it means

The epsilon-delta definition generalizes directly to vector inputs: no matter how small a tolerance ε around L we demand, some neighborhood of x_0 (of radius δ) guarantees the function stays within that tolerance.

Components

$B_{ϵ} (L)$ ball of radius ε around L in R
$B_{δ} (x_{0})$ ball of radius δ around x_0 in R^n

What it means

Unlike one-dimensional limits where checking left and right suffices, a limit in higher dimensions must hold along every possible path of approach, making existence much harder to establish (but easier to disprove).

What it means

Because a limit must agree along every path, finding just two directions with different limiting values is sufficient to conclude the overall limit fails to exist — though checking finitely many matching paths never proves existence.

What it means

When no indeterminate form arises, the limit of a multivariable elementary function can be computed by simple direct substitution.

What it means

As (x,y) approaches the origin, the denominator shrinks to 0 while staying positive, so the fraction grows without bound from every direction.

What it means

Since the exponential function is continuous, the limit can be pulled inside, reducing the problem to a limit inside the exponent, which diverges to -∞, giving overall limit 0.

What it means

By substituting t = xy, a two-variable indeterminate limit reduces to a well-known single-variable fundamental limit, which can then be substituted back.

What it means

Substituting t = x² + y² converts this two-variable indeterminate limit into the classic single-variable fundamental limit (eᵗ − 1)/t → 1.

What it means

This limit at (0,0) yields an indeterminate 0/0 form not reducible by substitution, so different paths must be tried; here the axes give 0 but the line y = x gives 1/4.

What it means

Because the axis approach and the diagonal approach give different values, the overall two-variable limit fails to exist.

What it means

Approaching along the line x = y gives 1, while approaching along x = -y gives -1; since these disagree, the limit does not exist.

What it means

By bounding the expression between 0 and a quantity that squeezes to 0 (using the fact that x²/(x²+y²) is always between 0 and 1), the squeeze theorem forces the original limit to also equal 0.

What it means

The definition of continuity extends unchanged to functions of several variables: the multivariable limit at a point must equal the function's actual value there. Informally: no holes and no jumps.

What it means

Just as in one variable, all standard building-block functions of several variables are continuous throughout their natural domains, and composing them preserves continuity.

What it means

Since this function is built from elementary pieces (a polynomial term plus a reciprocal), it is continuous everywhere it is defined, i.e. everywhere except where x = 0.

What it means

The min/max existence guarantee generalizes fully to higher dimensions: any continuous function on a compact subset of Rⁿ attains both a maximum and a minimum value.

Coercivity & Tonelli25

What it means

Weierstrass needs a compact domain, but in economics we usually only care about maximizers and often work on unbounded sets — so we look for weaker conditions that still guarantee a maximizer.

What it means

f is coercive on C when at least one of its upper contour sets meets C in a non-empty compact set. When C = A we simply say f is coercive.

Components

$(f \geq t)$ upper contour set at level t
$compact$ closed and bounded

What it means

One single level t = −2 with a non-empty compact upper contour set is enough for coercivity.

What it means

For k in (0,1] the upper contour set is a non-empty closed bounded interval, so e^{−|x|} is coercive even though it never diverges to −∞.

What it means

Every upper contour set of x² is unbounded, hence never compact, so x² is not coercive on ℝ (it has no maximum there).

What it means

Intersecting with a compact C makes the upper contour sets compact for k in (0,9], so x² is coercive on [−3,3].

What it means

The same function can be coercive on one set and not on another; you can often recover coercivity by restricting the domain.

What it means

Composing with a strictly increasing function changes neither coercivity nor the set of maximizers — the same trick used earlier in the course for quasi-concavity.

Components

$g \circ f$ the composite function x ↦ g(f(x))

What it means

A strictly increasing g preserves weak inequalities in both directions, so upper contour sets of f coincide with upper contour sets of g∘f at the transformed level g(k); compactness therefore transfers.

What it means

Decomposing h into a coercive inner function and a strictly increasing outer function proves coercivity without computing contour sets.

What it means

For a continuous function on a closed set, every upper (and lower) contour set intersected with C is closed — the key step toward coercivity on compact sets.

What it means

With C closed, every upper contour set is closed, as the lemma predicts.

What it means

If C is not closed the contour sets need not be closed — the closedness assumption in the lemma cannot be dropped.

What it means

On a compact set, continuity alone gives coercivity: contour sets are closed by the lemma and bounded because C is bounded.

What it means

The cubic illustrates all cases: unbounded contour sets kill coercivity, compactness of C guarantees it, and the last case shows the proposition gives sufficient but not necessary conditions.

What it means

Coercivity plus continuity guarantee a maximizer, even when C is unbounded — a strict weakening of Weierstrass for maximization problems.

Components

$max_{y \in C} f (y)$ the maximum value of f on C

What it means

The compact upper contour set isolates a compact piece of the domain that still contains all the high values of f, so Weierstrass applies there and the max found is the global max on C.

What it means

Apply Weierstrass on the compact upper contour set; every point outside it has a value below the level t, which is already dominated, so the local maximizer is global on C.

What it means

Economic and financial optimization problems often have unbounded choice sets; Tonelli gives existence of an optimum under conditions that such problems typically satisfy.

What it means

f must diverge to −∞ along every unbounded sequence: as points get far from the origin in any direction, the value becomes arbitrarily negative.

Components

$∥ x_{n} ∥$ Euclidean norm of the n-th term

What it means

The value falls to −∞ in every direction, so every unbounded sequence sends f to −∞.

What it means

Along the diagonal the function stays constant, so it fails to diverge to −∞ on that unbounded sequence.

What it means

e^{−|x|} is coercive but not supercoercive — supercoercivity is strictly stronger.

What it means

Supercoercivity is exactly boundedness of all upper contour sets: if some (f ≥ k) were unbounded we could pick a sequence inside it with norms diverging while f(xₙ) ≥ k.

What it means

Bounded contour sets (supercoercivity) plus closed contour sets (continuity on a closed C) give compactness at every level, hence coercivity and — with Tonelli — a maximizer on C.

Symbol index

Every symbol used in the sheet, and what it stands for.

$A, B$ sets, denoted by capital letters
$x, a$ elements, denoted by lowercase letters
${\cdot}$ set notation
$x ∣ P (x)$ the set of x satisfying property P
$\in$ is an element of
$\in /$ is not an element of
$\subseteq$ subset or equal
$\subset$ proper subset
$\exists$ there exists
$⊈$ is not a subset of
$A = B$ set equality
$\emptyset$ the empty set
$\cup$ union operator
$\cap$ intersection operator
$A \cap B = \emptyset$ condition for disjointness
$⋃_{i = 1}^{N} A_{i}$ union over the family of sets A_1,...,A_N
$⋂_{i = 1}^{N} A_{i}$ intersection over the family of sets A_1,...,A_N
$∖$ set difference operator
$\cup, \cap$ union and intersection
$U$ the universal set
$A^{c}$ complement of A
$A ∖ B$ set difference
$B^{c}$ complement of B
$(A^{c})^{c}$ complement of the complement of A
$(A \cap B)^{c}$ complement of the intersection
$A^{c} \cup B^{c}$ union of the complements
$N$ set of natural numbers
$Z$ set of integers
$Q$ set of rational numbers
$R$ set of real numbers
$Q^{c}$ irrational numbers
$p, q, r$ real/rational numbers
$>, <, =$ order relations
$\geq, \leq$ weak order relations
$A$ subset of R
${z ∣ x \leq z \leq y}$ all points between x and y
$[x, y]$ closed interval
$(x, y)$ open interval
$[x, y), (x, y]$ half-open intervals
$\infty, - \infty$ positive/negative infinity, used as unbounded endpoints
$R$ extended real line
$h$ candidate bound
$max A, min A$ maximum and minimum of A
$sup A$ least upper bound of A
$in f A$ greatest lower bound of A
$sup A, max A$ supremum and maximum of A
$in f A, min A$ infimum and minimum of A
$∣ x ∣$ absolute value of x
$∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ triangle inequality
$c$ positive constant
$d (x, y)$ distance between x and y
$B_{ϵ} (x_{0})$ neighborhood of x_0 with radius epsilon
$ϵ$ radius
$x_{0}$ center point
$B_{ϵ}^{+} (x_{0})$ right half-neighborhood
$B_{ϵ}^{-} (x_{0})$ left half-neighborhood
$B_{x_{0}} (\infty), B_{x_{0}} (- \infty)$ one-sided infinite neighborhoods
$B_{ϵ}^{-} (x)$ left neighborhood of x with radius epsilon
$B_{ϵ}^{+} (x)$ right neighborhood of x with radius epsilon
$\times$ Cartesian product operator
$(a_{1}, a_{2})$ ordered pair
$R^{2}$ the Cartesian plane
$(12)$ column vector representation of the point (1,2)
$R^{3}$ three-dimensional Cartesian space
$(x, y, z)$ ordered triple
$R^{n}$ n-dimensional Cartesian space
$(x_{1}, \dots, x_{n})$ ordered n-tuple
$x_{1}, \dots, x_{n}$ components of the vector
$x, y \in R^{n}$ vectors of the same dimension n
$x_{i}, y_{i}$ i-th components of x and y
$u, v$ two vectors sharing a starting point
$x, y, z \in R^{n}$ arbitrary vectors
$0$ zero vector
$- x$ additive inverse of x
$α \in R$ scalar
$x \in R^{n}$ vector
$x, y \in R^{n}$ vectors
$α, β \in R$ scalars
$α_{i}$ scalar coefficients
$x_{i}$ vectors being combined
$x \cdot y$ resulting scalar
$x ⊥ y$ x is orthogonal to y
$α \in [0, 1]$ weight parameter
$α_{i} \geq 0$ non-negative weights
$\sum α_{i} = 1$ weights sum to one
$A \subseteq R^{n}$ the set in question
$∥ x ∥$ length (norm) of x
$ϵ > 0$ radius
$x_{0} \in R^{n}$ center point
$d (x, x_{0})$ Euclidean distance from x to x0
$int A$ set of interior points of A
$ext A$ set of exterior points of A
$\partial A$ set of boundary points of A
$A = cl A$ closure of A
$A^{'}$ derived set of A (set of all limit points)
$int A$ interior of A
$B_{ϵ} (x)$ open ball of radius ϵ around x
$K$ a fixed positive radius bounding all points of A
$B_{K} (0)$ ball of radius K centered at the origin containing A
$δ$ radius of the smaller ball around y, chosen as ϵ − ‖x−y‖
$B$ codomain of f
$f (x) = y$ f associates y ∈ B to x ∈ A
$x$ independent variable / input
$y = f (x)$ dependent variable / output, the image of x
$Im (f)$ range / image of f
$Gr (f)$ graph of f
$m$ slope
$b$ y-intercept
$a, b, c$ real coefficients, a ≠ 0
$k$ proportionality constant
$α$ exponent, α ∈ ℚ
$a$ vertical shift amount
$f$ inner function, f: A → B
$g$ outer function, g: C → D, with f(A) ⊆ C
$f^{- 1}$ inverse function of f
$f : A \subseteq R \to R$ the function under consideration
$Im (f)$ the image (range) of f
$f : R^{n} \to R$ a multivariable real-valued function
$sup_{x \in A} f (x)$ least upper bound of the function's values on A
$in f_{x \in A} f (x)$ greatest lower bound of the function's values on A
$e^{2}$ supremum, attained at x = 2
$f increasing$ hypothesis: x>y \Rightarrow f(x)\ge f(y)
$max_{x \in A} f (x)$ the maximum of Im(f), i.e. the global maximum value
$C$ the set over which we optimize
$arg max$ set of maximizers
$K, L$ capital and labor inputs
$p_{K}, p_{L}$ prices of capital and labor
$λ$ Lagrange multiplier on the budget constraint
$w, c$ quantities of wine and cheese consumed
$B_{ϵ}^{+} (x)$ right-neighborhood of x of radius epsilon
$B_{δ}^{-} (x)$ left-neighborhood of x of radius delta
$α f (x) + (1 - α) f (y)$ value of the chord (line segment) at that point
$epi (f)$ the set of points lying on or above the graph of f
$hypo (f)$ the set of points lying on or below the graph of f
$g_{i}$ single-variable function acting only on coordinate x_i
$x = (x_{1}, \dots, x_{n})$ point in the domain A
$f (x)$ global maximum/minimum value
$C \subseteq A$ subset of the domain over which we optimize
$α x + (1 - α) y$ convex combination of x and y
$ω_{i} \geq 0$ non-negative weights
$f^{- 1} (k)$ preimage of the value k, i.e. the level set
$u$ utility function
$(u \geq k)$ bundles delivering at least utility level k
$p_{x}, p_{y}$ prices of the two goods
$w$ consumer's wealth/budget
$g \circ f$ composition of increasing transformation g with f
$x_{n}$ the n-th term of the sequence
$f : N \to R$ sequence viewed as a function
$Im (f)$ range: the set of terms
$L$ the limit of the sequence
$n_{ϵ}$ threshold index depending on ε, beyond which terms are within ε of L
$B_{ϵ} (L)$ epsilon-neighborhood (open ball) of L
$B_{k} (\infty)$ neighborhood of infinity: the interval (k, ∞)
$B_{k} (- \infty)$ neighborhood of negative infinity: the interval (−∞, k)
$L^{-}$ notation indicating convergence to L from below
$L^{+}$ notation indicating convergence to L from above
$d (a_{n}, L)$ distance between the term aₙ and the limit L, i.e. |aₙ − L|
$x_{n} > 0 eventually$ the terms are positive from some index onward
$n^{+}$ threshold index beyond which xₙ has the same sign as L
$sup {x_{n}}$ least upper bound of the sequence, which exists by completeness of R when bounded above
$n^{α}$ power sequence with exponent α
$q^{n}$ exponential sequence with ratio q
$(ln n)^{α}$ logarithmic sequence with exponent α
$q$ common ratio
$L, H$ limits of the two sequences
$y_{n} \leq x_{n} \leq z_{n}$ squeeze inequality, holding eventually (for large n)
$\frac{x _{n + 1}}{x _{n}}$ ratio of consecutive terms' absolute values
$L < 1$ limit of the ratio, strictly less than 1
$O (b_{n})$ little-o notation: negligible with respect to b_n
$a_{n} ≍ b_{n}$ a_n and b_n are of the same order (comparable)
$a_{n} \sim b_{n}$ a_n asymptotically equivalent to b_n
$e$ Euler's/Napier's number, approximately 2.71828
$x_{n} \in R^{m}$ the n-th vector term, with m components
$∥ x_{n} - L ∥$ Euclidean distance between x_n and L
$a_{n}$ general term of the sequence
$S$ purported sum of the series
$(\frac{1}{2})^{n}$ distance walked at step n
$\sum_{n = 0}^{\infty} a_{n}$ the series generated by aₙ
$s_{n}$ n-th partial sum
$s_{n} = \frac{1 - q ^{n + 1}}{1 - q}$ closed form of the n-th partial sum for q ≠ 1
$n_{0}$ starting index of summation
$a q^{n_{0}}$ first term actually appearing in the sum
$regular$ either convergent or divergent to +infinity
$α, β$ exponent parameters
$\frac{a _{n + 1}}{a _{n}}$ ratio of consecutive terms
$e^{x}$ the exponential function evaluated at x
$x = 1$ point excluded from the domain but still a limit point of it
$d (x, y) = ∣ x - y ∣$ distance on the real line
$B_{δ} (x_{0})$ neighborhood of x_0 (possibly infinite)
$M$ arbitrarily large target bound for f(x)
$lim_{x \to x_{0}} f (x)$ value the function approaches near x_0
$x_{n} \to x_{0}, x_{n} \neq = x_{0}$ sequence approaching x_0 but never equal to it
$A compact$ A is closed and bounded
$max_{z \in A} f (z)$ the maximum value attained by f on A
$p_{w}, p_{c}$ strictly positive prices of wine and cheese
$f (a) \cdot f (b) \leq 0$ f(a) and f(b) have opposite signs (or one is zero)
$m, M$ minimum and maximum values of f on [a,b] (exist by Weierstrass)
$(f \geq t)$ upper contour set at level t
$compact$ closed and bounded
$max_{y \in C} f (y)$ the maximum value of f on C
$∥ x_{n} ∥$ Euclidean norm of the n-th term

Sets & set operations35

What is a Set?

Ways to Describe a Set

Set Membership

Subset

Proper Subset

Not a Subset

Set Equality

Empty Set

Union

Intersection

Disjoint Sets

Union and Intersection of Families of Sets

Set Difference

Commutative Property of Union and Intersection

Associative Property of Union and Intersection

Distributive Property of Set Operations

Universal Set

Complement

Set Difference via Complement

Double Complement

De Morgan's Laws

Proof Sketch of De Morgan's Law (A∩B)ᶜ = Aᶜ∪Bᶜ

Natural Numbers

Integers

Rational Numbers

Proof that √2 is Irrational

The Real Numbers

Hierarchy of Number Sets

Density and Continuity of Q and R

The Natural Order on R

R² Has No Natural Total Order

Interval

Types of Bounded Intervals

Unbounded Intervals

The Extended Real Line

Real numbers & completeness16

Upper and Lower Bound

Bounded Above / Below / Bounded

Maximum and Minimum

Uniqueness of Maximum and Minimum

Supremum and Infimum

Relating Max/Min to Sup/Inf

Completeness of R

Absolute Value

Properties of the Absolute Value

Absolute Value Inequality

Distance in R

Neighborhood (Open Ball) in R

Half Neighborhoods

Infinite Neighborhoods

Characterization of the Supremum

Characterization of the Infimum

Vectors & Rⁿ34

Cartesian Product

R²: The Cartesian Plane

Vectors in R²

R³: Cartesian 3-Space

Rⁿ: General Cartesian Space

Column and Row Vector Notation

Vector Addition

Vector Addition Requires Matching Dimension

Parallelogram Rule

Properties of Vector Addition

Scalar Multiplication

Properties of Scalar Multiplication

Vector Space

Linear Combination

Inner (Dot/Scalar) Product

Geometric Meaning of Inner Product Sign

Orthogonal Vectors and Orthogonal Sets

Properties of the Inner Product

Order on Rⁿ: Equality

Order on Rⁿ: Weak Inequality

Order on Rⁿ: Strict Inequality

Order on Rⁿ: Strong Inequality

Not Comparable

Relationships Between Order Notions and Partial Order

Positivity Notions

Norm in R (Absolute Value)