Alphabeta Math

Sequences and Series of Functions; Uniform Convergence

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 15 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 15 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions

Lemma

Uniform convergence of real-valued functions implies pointwise convergence

Theorem

A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy

Definition

A series of real-valued functions and its pointwise and uniform convergence through its partial sums

Corollary

A series of real-valued functions converges uniformly if and only if its tails are uniformly small

Lemma

Uniform limits respect sums and scalar multiples

Lemma

Products converge uniformly when both factors converge uniformly and one limiting factor and one approximating family are uniformly bounded

Theorem

The uniform limit of continuous real-valued functions on a metric space is continuous

Corollary

The uniform limit of uniformly continuous real-valued functions is uniformly continuous

Definition

The space C(K,ℝ) of continuous real-valued functions on a nonempty compact metric space

Theorem

C(K,ℝ) is complete in the supremum metric for every nonempty compact metric space K

Lemma

Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error

Theorem

A uniform limit of Riemann-integrable functions is Riemann integrable, and its integral is the limit of their integrals

Theorem

If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit

Theorem

The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series

Theorem

Uniform Dirichlet test: uniformly bounded partial sums times a uniformly decreasing null family give a uniformly convergent function series

Theorem

Uniform Abel test: a uniformly convergent function series times a uniformly bounded pointwise monotone family gives a uniformly convergent product series

Theorem

Dini's theorem on a closed interval: monotone pointwise convergence of continuous functions to a continuous limit is uniform

Remark

Agreement of the quantified real-valued definition with the later uniform-metric and uniform-topology formulations

Sources