ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Montel's diagonal extraction can be written out concretely on a disc

Example

Assume the Axiom of Choice.

On the unit disc, Montel's diagonal extraction can be written concretely by the compact discs $K_{n} : = {z : ∣ z ∣ \leq 1 - 1 / (n + 1)} .$ Given any locally bounded sequence in $H (D)$ , one may choose a subsequence converging uniformly on each $K_{n}$ , and the diagonal subsequence then converges locally uniformly on all of $D$ .

Facts & Assumptions

Given: Choice and a locally bounded sequence $(f_{j})$ in $H (D)$ .

[L1]

Montel's theorem supplies a uniformly convergent subsequence on each compact stage of the canonical exhaustion (Montel's theorem: every locally bounded holomorphic family is normal).

Verification

technique · direct

1.1

Apply [L1] to choose a subsequence converging uniformly on the first compact disc, then a further subsequence converging uniformly on the second compact disc, and continue stage by stage.

L1givenchoose

2.1

The diagonal term at stage $n$ lies in every earlier chosen subsequence, so for each fixed compact stage the diagonal sequence eventually belongs to the corresponding uniformly convergent subsequence. Hence the diagonal sequence converges locally uniformly on the whole disc.

given∎

Depends on

Montel's theorem: every locally bounded holomorphic family is normal

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)