LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

The uniform-convergence uniformity is finer than the pointwise uniformity, and they agree when the domain is finite

Statement

The uniform-convergence uniformity on $Y^{X}$ is finer than the pointwise-convergence uniformity. If $X$ is finite, they are equal.

Facts & Assumptions

Given: A uniform space $Y$ , a set $X$ , an entourage $V$ , and finite $F \subseteq X$ .

[L1]

Pointwise basic entourages require $V$ -closeness on $F$ , while uniform basic entourages require it on all of $X$ (The pointwise and uniform-convergence uniformities on a function set $Y^{X}$ ).

[L2]

Finiteness allows $X$ itself as an allowed finite coordinate set (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1

$Q (V) \subseteq P (F, V)$ , so every pointwise basic entourage contains a uniform basic entourage.

1.2

If $X$ is finite, $P (X, V) = Q (V)$ by [L1] and [L2], so each uniform basic entourage is pointwise basic as well.

L1L2

2.1

Hence uniform convergence is finer than pointwise convergence.

step 1.1

3.1

The two uniformities are equal in the finite-domain case.

step 2.1step 1.2∎

Depends on

Used by

The functions fₙ(k)=1 for k≥ n and 0 otherwise converge pointwise but not uniformly on ℕ Example

Dependency tree · two levels

10 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

M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)