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Every pointwise-bounded equicontinuous sequence in has a uniformly convergent subsequence
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()) and the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain). Let be a nonempty compact metric space. Every equicontinuous pointwise-bounded sequence in has a subsequence converging uniformly to a member of .
Facts & Assumptions
Given: A nonempty compact metric space , the stated choice principles, and an equicontinuous pointwise-bounded sequence in .
For a nonempty compact metric space , an equicontinuous pointwise-bounded family in has compact closure in the supremum metric (Arzelà--Ascoli for real under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded).
Assuming Countable Choice and Dependent Choice, a 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
The sequence lies in its compact closure by [L1].
By [L2], it has a subsequence converging in the supremum metric to a point of that closure.
Supremum-metric convergence is uniform convergence, so the claimed subsequence converges uniformly.
Depends on
- Arzelà--Ascoli for real $C(K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- 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
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Dependency tree · two levels
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Sources
- The Ascoli--Arzelà Theorem (MIT) (standard reference, not scraped)