LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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.

Uniform convergence of real-valued functions implies pointwise convergence

Statement

Let $X$ be a set. If a sequence of functions $f_{k} : X \to R$ converges uniformly to $f : X \to R$ , then it converges pointwise to $f$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

Facts & Assumptions

Given: A set $X$ , functions $f_{k}, f : X \to R$ , and uniform convergence $f_{k} \to f$ on $X$ .

[A1]

Uniform convergence means that for every real $ε > 0$ there is $N \in N$ such that $∣ f_{k} (x) - f (x) ∣ < ε$ for every $k \geq N$ and every $x \in X$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

Proof

technique · direct

1.1

Fix $x \in X$ and a real $ε > 0$ . By [A1] choose $N \in N$ such that $∣ f_{k} (y) - f (y) ∣ < ε$ for every $k \geq N$ and every $y \in X$ .

A1choose

2.1

In particular, $∣ f_{k} (x) - f (x) ∣ < ε$ for every $k \geq N$ .

step 1.1

3.1

Since $x$ and $ε$ were arbitrary, $f_{k} (x) \to f (x)$ for every $x \in X$ , which is pointwise convergence.

step 2.1A1∎

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I, §6.1 (standard reference, not scraped)
W. Trench, Introduction to Real Analysis (standard reference, not scraped)