proposition¶
A statement that is either true or false — proof is the art of combining propositions with connectives and tracking how the truth values are forced.
Key terms from the scenes, grouped by chapter. Each term links to the scene where it is introduced.
A statement that is either true or false — proof is the art of combining propositions with connectives and tracking how the truth values are forced.
A proof that takes the hypothesis as given and reaches the conclusion by a straight chain of justified steps — no detours.
Splitting the domain into cases that cover everything, and proving the claim in each case — together the cases cover the whole proof.
— apply first, then to the result. Injectivity and surjectivity propagate through composition in provable ways.
(for all) and (there exists) — the symbols that pin down precisely how a statement ranges over its domain. Negation swaps the two.
The contrapositive of is — always equivalent to the original, so it can be proved in its place when the direct route stalls.
The method that assumes the negation of the conclusion and derives a contradiction — if assuming yields a contradiction (), then holds.
— an "if and only if" claim that is completed only by proving the two directions and separately.
Induction that assumes all of when proving — used when the step needs much smaller values, like the factors of .
A relation that is reflexive, symmetric, and transitive — its equivalence classes partition the whole set without overlap.
— 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 .
The with ; its existence is guaranteed by .
The axiom that a choice function exists for any collection of nonempty sets.
If every chain has an upper bound, a maximal element exists. Equivalent to the axiom of choice.
Every set is smaller than its power set: , proved by diagonalization.
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.
No rational satisfies — proved by infinite descent on lowest terms, exposing a "hole" in the rational line.
Between any reals lies a rational — take and , then .
A number of the form with and — the same object as the plane point .
The modulus of — the distance from the origin to the point , satisfying the triangle inequality .
The conjugate of is — reflection across the real axis, satisfying .
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 .
An array of numbers indexed by two indices .
The Pringsheim limit: when for every there is with whenever . Both indices grow together.
A limit taken in one index first and the other second, or . Swapping the order can change the value, or one side may fail to exist.
If then ; if then — the basic fates of powers and roots.
() — Cauchy–Schwarz is .
— the identity every inner-product norm satisfies. A norm that violates it is induced by no inner product.
In the real case, — the formula that reads the inner product back off the norm alone. The norm completely determines the inner product.
The theorem that a norm is induced by an inner product — the one given by the polarization identity — if and only if it satisfies the parallelogram law.
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.
The - definition, sequential preservation, and open preimages of open sets — three lenses on the same continuity.
(reduced), (irrational) — the popcorn function, continuous at every irrational and discontinuous at every rational.
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 set that is both open and closed. In only and qualify, but subspaces admit nontrivial examples.
The closure of — the canonical example of a connected but not path-connected curve.
A continuous surjection — a space-filling curve. Injectivity is impossible.
A set that cannot be split into two disjoint nonempty open sets. is the prototype; fails completely.
A decomposition with open in , disjoint, and both nonempty. If a separation exists, is disconnected.
A space in which every subset with at least two points is disconnected — the only connected subsets are singletons and . is the classic example: an irrational cut always slips between two rationals.
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 .
For a decreasing positive function, the series and the improper integral converge or diverge together.
A series with terms on the grid ((m,n)); with absolute convergence, rows-first and columns-first sums agree.
: 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 bounded continuous functions on with the sup norm . It is complete.
The continuous functions vanishing at infinity: for every , outside some compact set. A closed subspace of , hence complete.
The continuous functions with compact support. Dense in but neither complete nor closed: .
The limiting Gibbs overshoot — about 8.949% of the jump. is the sine integral.
Fourier coefficients of an integrable function tend to 0 — rapid oscillation averages every bump away.
— conservation of energy.
Having a tangent line at the point; differentiability implies continuity.
If is continuous on and differentiable on , some satisfies .
If , has a local inverse near with .
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.
makes increasing, strictly increasing (by MVT) — the sign of the derivative decides rise and fall.
is differentiable everywhere, yet its derivative is discontinuous at — differentiable does not imply continuously differentiable.
; 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 product rule, integrated.
With , : the limits follow the substitution.
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.
1 on rationals, 0 on irrationals — every upper sum is 1 and every lower sum 0, so it is not Riemann integrable.
— reduces to the Riemann integral for ; a jump lets one point's value dominate.
Monotone convergence, Fatou, dominated convergence — three conditions for . Escaping mass breaks them.
For nonnegative (Tonelli) or integrable (Fubini) functions, iterated integrals agree in either order.
On finite measure, a.e. convergence is uniform after removing a set of arbitrarily small measure.
If , the limsup set has measure 0; with independence the converse holds.
The solid of revolution of () — finite volume , infinite surface area. The painter's paradox.
A choice-set of representatives of via the axiom of choice — not Lebesgue measurable.
— holds when is continuous. Also called Leibniz's rule.
The triangle inequality () — derived from Hölder's inequality.
A curve whose polygonal lengths have a finite supremum, — that supremum is its length. Continuity alone is not enough: the graph of has infinite length.
The length of a curve . When is , this equals .
The broken line joining the curve's values at partition points, and its length — refinement never shortens it, by the triangle inequality.
— flip one function, slide it, and take the weighted overlap with the other; the operation commutes.
A family of kernels of mass concentrating near the origin — at continuity points, so convolution approaches the identity operation.
The support of a function is the closure of — the region where is nonzero, closed up.