CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Under the Axiom of Choice, a pointwise bounded equicontinuous sequence on a nonempty compact metric domain into a proper metric target has a uniformly convergent subsequence

Statement

Assume the Axiom of Choice. Let $X$ be a nonempty compact metric space, let $Y$ be a proper metric space, and let $(f_{k})_{k \in N}$ be an equicontinuous sequence in $C (X, Y)$ . If ${f_{k} (x) : k \in N}$ is bounded for every $x \in X$ , then some subsequence converges uniformly to a member of $C (X, Y)$ .

Facts & Assumptions

Given: Choice, a nonempty compact metric space $X$ , a proper metric space $Y$ , and a pointwise bounded equicontinuous sequence $(f_{k})$ .

[L1]

Equicontinuity and compact coordinate closures make the uniform closure compact (Ascoli–Arzelà in the uniform topology for nonempty compact metric domains).

[L2]

Under Countable Choice and Dependent Choice, every compact metric space is sequentially compact (For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice).

Proof

technique · direct

1.1

For each $x \in X$ , the coordinate set ${f_{k} (x) : k \in N}$ is bounded. Its closure is closed and remains bounded, and therefore is compact by properness.

given

2.1

By equicontinuity, step 1.1, and [L1], the uniform closure $H$ of ${f_{k} : k \in N}$ is compact.

L1step 1.1

3.1

Choice implies the two weaker choice principles in [L2], so the compact metric space $H$ is sequentially compact. Hence $(f_{k})$ has a subsequence converging in the uniform metric to some $f \in H \subseteq C (X, Y)$ .

L2step 2.1choose

4.1

Convergence in the uniform metric is uniform convergence, which gives the claimed subsequence and continuous limit.

step 3.1∎

Depends on

Used by

Real and finite-dimensional Euclidean Ascoli–Arzelà criteria Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 80 results over 18 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

The Ascoli–Arzelà Theorem, BBT (standard reference, not scraped)
The Arzelà–Ascoli Theorem (standard reference, not scraped)