LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $f_k:X\to\mathbb{R}$ converges uniformly to $f:X\to\mathbb{R}$ , 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 $f_k,f:X\to\mathbb{R}$ , and uniform convergence $f_k\to f$ on $X$ .

[A1]

Uniform convergence means that for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that $|f_k(x)-f(x)|<\varepsilon$ for every $k\ge N$ 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 $\varepsilon>0$ . By [A1] choose $N\in\mathbb{N}$ such that $|f_k(y)-f(y)|<\varepsilon$ for every $k\ge N$ and every $y\in X$ .

A1choose

2.1

In particular, $|f_k(x)-f(x)|<\varepsilon$ for every $k\ge N$ .

step 1.1

3.1

Since $x$ and $\varepsilon$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)