natural number
— the basic unit of counting, axiomatized by Peano.
Key terms from the scenes, grouped by chapter. Each term links to the scene where it is introduced.
— the basic unit of counting, axiomatized by Peano.
The proof principle getting for every natural from and — like dominoes falling in turn.
The five axioms defining the naturals — existence of , the successor function, the induction axiom, and more. Addition and multiplication are defined recursively from them.
— the number of ways to choose from . Satisfies Pascal's identity .
— proved by induction, with Pascal's identity as the inductive step.
— built as equivalence classes of natural pairs with (the number "").
() — built as equivalence classes of integer pairs with ; they form a field.
The set of elements equivalent under an equivalence relation — the key technique for "building" the integers, rationals, and reals.
A collection of distinct objects. Every discussion in analysis begins with a set.
A rule assigning to each element of a domain exactly one element of a codomain.
The least of the upper bounds of a set, written .
The greatest of the lower bounds of a set, written .
A set that can be put in one-to-one correspondence with the natural numbers; the rationals are the classic example.
A set with one associative operation, an identity element, and inverses.
A set forming a group under each of addition and multiplication, with distributivity; the rationals and the reals are examples.
A set closed under addition and scalar multiplication; is the prototype.
A structure-preserving bijection. Isomorphic spaces are "the same structure in different packaging".
The number of elements in a basis of a vector space.
The dimension of the image of a linear map.
when for every there is with for all .
The "length" of a vector; the metric is .
The space of sequences with finite -th power sum; .
— the inner product never exceeds the product of the norms; equality holds exactly for parallel vectors.
A real number that is not rational; is the classic example.
An equivalence class of Cauchy sequences of rationals — the rationals with their gaps filled.
A sequence whose terms get arbitrarily close to each other; equivalent to convergence in .
Every bounded real sequence has a convergent subsequence.
A sequence that is always increasing or always decreasing; if bounded, it converges.
The value the tail suprema \\\sup_{n \\ge N} a_n\ settle to as \N \\to \\infty\ — the largest cluster point of the sequence.
The value the tail infima \\\inf_{n \\ge N} a_n\ settle to as \N \\to \\infty\ — the smallest cluster point of the sequence.
The sequence l'Hôpital: if is strictly increasing and unbounded and the difference ratio tends to , so does .
The rigorous definition of a function limit: for every error there is a tolerance .
as ; the graph has no breaks.
Every discontinuity of a monotone function is a jump; only finitely many jumps exceed , so the discontinuities are at most countable.
Continuity with a single working across the whole domain; is continuous but not uniformly continuous on .
A continuous function takes every value between any two of its values.
A continuous function on a closed interval attains a maximum and a minimum.
Convergence of a sequence of functions with one working for all ; it preserves continuity of the limit.
Convergence at each point separately; the limit may be discontinuous.
A set in which every point is an interior point; it contains none of its boundary.
A set whose complement is open; it contains all its limit points.
A set together with all its limit points, written .
A set that fits inside some interval (or ball).
A closed and bounded set (Heine–Borel in ); every sequence has a subsequence converging inside the set.
A set of reals is compact if and only if it is closed and bounded.
A set equipped with a distance between points.
A distance function satisfying symmetry, non-negativity, and the triangle inequality.
A metric space where every Cauchy sequence converges to a point of the space.
A point with ; a contraction on a complete metric space has a unique one.
A countable union of nowhere dense sets — by the Baire Category Theorem, a complete metric space cannot be meagre in itself.
A topological space in which every countable intersection of dense open sets is dense — complete metric spaces and locally compact Hausdorff spaces are the key examples.
A function expressible as the pointwise limit of a sequence of continuous functions — Thomae's function is the poster child; its continuity points always contain a dense .
At , measures local spread; iff is continuous at .
The set left after repeatedly removing middle thirds; it has length zero yet is uncountable.
A Cantor-type set built by removing only length at each stage — compact, nowhere dense and perfect, yet keeping measure . Also called the Smith–Volterra–Cantor set.
The continuous nondecreasing function that climbs only on the Cantor set — constant on every complementary interval, so on a set of total length .
Nickname for the Cantor function — infinitely many flat steps, yet it climbs from 0 to 1.
The class of functions for which the FTC holds — guaranteeing . The Cantor function is continuous but not absolutely continuous.
A family of functions that is continuous with "the same " at every point — for each there is working for all . The key hypothesis of the Arzelà–Ascoli theorem.
A set whose closure is compact — every sequence hides a convergent subsequence. In this is equivalent to uniform boundedness + equicontinuity (Arzelà–Ascoli).
Every continuous function on a closed interval can be uniformly approximated by polynomials.
; used in the constructive proof of the approximation theorem.
The property that for any two distinct points there is a function in the algebra with . One of the key hypotheses of the Stone–Weierstrass theorem.
A set of functions closed under linear combinations and products. Trigonometric polynomials form a subalgebra via product-to-sum identities (e.g. ), and by the Stone–Weierstrass theorem it is uniformly dense in .
For , is dense in if and only if .
: its terms tend to 0, yet the sum diverges.
A series whose partial sums have no finite limit.
— the value the gap between harmonic numbers and the logarithm settles at.
— the partial sums of the harmonic series.
The problem of showing .
For , — sieving out composite terms leaves only the product over primes.
A series with alternating signs; it converges if its terms decrease monotonically to 0 (Leibniz test).
Deciding convergence by comparing terms against a reference series.
Deciding convergence from the limit of .
: converges if , diverges if . The test to reach for when the ratio drifts to 1 and the ratio test goes silent.
With : convergence if stays positive, divergence if and diverges. gives the ratio test, gives Raabe's.
If : converges for , diverges for — a corollary of Kummer's test.
For decreasing , and share convergence.
A series whose neighboring terms cancel, leaving a simple partial sum.
; converges to when .
An infinite product representation of .
The asymptotic .
.
; divergent in the usual sense.
The limit of the averaged partial sums; the Grandi series sums to this way.
If the partial sums of \(a_n)\ are bounded and \(b_n)\ decreases to 0, then \\\sum a_n b_n\ converges.
If converges and is monotone and bounded, then converges. Unlike Dirichlet's test, need not tend to 0.
— the discrete integration by parts; the engine behind Abel's test.
A series of the form .
The radius of the interval on which a power series converges.
A sparse layout of exponents in with — e.g. (ratio exactly 2). With radius 1, the whole unit circle becomes a natural boundary.
A curve past no point of which a holomorphic function can be analytically continued — for the unit circle is exactly that, by the Hadamard gap theorem.
If then — the boundary value of the power series recovers the ordinary sum.
The value of when it exists. Assigns values to divergent series, but is never "the sum" — the Abel sum of Grandi's series is .
A theorem recovering ordinary convergence from a weaker summability method (like Abel summation) under an extra condition on the terms, e.g. — Tauber's theorem is the prototype.
The base of the natural logarithm: .
; the Basel problem is .
Absolutely convergent series keep their sum under reordering; conditionally convergent ones need not.
Convergence without absolute convergence.
Convergence of ; it implies convergence.
A conditionally convergent real series can be rearranged to converge to any real number (or ) — because its positive and negative parts each diverge.
The product of two series with — diagonal sums.
If one series converges absolutely and the other converges, their Cauchy product converges to the product of the sums.
If \|f_n(x)| \\le M_n\ for all \x\ and \\\sum M_n\ converges, then \\\sum f_n\ converges uniformly.
On a compact set, a monotone sequence of continuous functions converging pointwise to a continuous limit converges uniformly.
A periodic function expressed as a sine-cosine series.
— a nonnegative trigonometric polynomial of mean 1 that turns Cesàro means of Fourier sums into a convolution.
The average of partial sums — it can sum divergent series, and for Fourier series it smooths out the ringing (Fejér's theorem).
Fourier partial sums overshoot by about 8.95% of the jump near a discontinuity — raising narrows the bump but never removes it.
The limiting Gibbs overshoot — about 8.949% of the jump. is the sine integral.
Having a tangent line at the point; differentiability implies continuity.
If is continuous on and differentiable on , some satisfies .
The mean value theorem for two functions; the basis of l'Hôpital's rule.
A or limit equals the limit of the differentiated numerator over denominator.
A derivative has the intermediate value property even when discontinuous — if then some has .
The intermediate value property without continuity — derivatives are the canonical examples.
An interior local max/min where is differentiable forces . Used to prove Darboux's theorem.
0 for , 1 for . Its jump discontinuity means it can never be anyone's derivative.
Left and right limits both exist but differ. Impossible in a derivative.
Continuous everywhere but differentiable nowhere — . Shows a uniform limit of smooth functions need not be differentiable.
The power series built from a function's derivatives at .
The theorem giving the Taylor polynomial together with the exact form of the remainder .
A function whose chords lie above its graph: .
For convex : of the average the average of .
The iteration for finding roots.
A contraction on a complete metric space has a unique fixed point.
A map that shrinks distances by a fixed ratio below 1.
; the continuous extension of the factorial ().
. Linked to the gamma function by .
The property that is convex. Together with and , it characterizes uniquely (Bohr–Mollerup theorem).
An integral over an unbounded interval or of an unbounded function, defined as a limit.
The limit of Riemann sums over finer and finer partitions.
The set of cut points dividing an interval.
Differentiable everywhere with a bounded derivative, yet the derivative is not Riemann integrable — built by stacking -type bumps over a fat Cantor set.
Differentiation and integration are inverse operations.
The sum stays under a finite ceiling as partitions refine. Every monotone function has bounded variation.
A function of bounded variation is the difference of two increasing functions: .
Writing a BV function as the difference of two increasing functions — always possible with (the variation function) and .
: the total variation of over . For a BV it is increasing, and is increasing too — giving the Jordan decomposition .
The extension of length, area, and volume to sets.
A set coverable by intervals of arbitrarily small total length.
Integration by slicing the range of values; applies to a wider class of functions than the Riemann integral.
The property that for every there is with whenever — the "absolute" strengthening of uniform continuity. Exactly the indefinite integrals of functions.