ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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.

A probability-convergent sequence with a prescribed fast almost-sure subsequence

Example

Suppose $X_{n} \to X$ in probability. Given any sequences $a_{k} > 0$ with $a_{k} ↓ 0$ and $b_{k} > 0$ with $\sum_{k} b_{k} < \infty$ , one can choose increasing $n_{k}$ so that $P (∣ X_{n_{k}} - X ∣ > a_{k}) < b_{k}$ and $X_{n_{k}} \to X$ almost surely.

Facts & Assumptions

Given: $X_{n} \to X$ in probability, $a_{k} > 0$ with $a_{k} ↓ 0$ , and $b_{k} > 0$ with $\sum_{k} b_{k} < \infty$ .

[L1]

Probability convergence supplies a later index for every positive threshold and positive bound (Convergence in probability).

[L2]

The least-index construction with a summable error schedule yields an almost-surely convergent subsequence (An almost-surely convergent subsequence from convergence in probability).

Verification

technique · constructive

1.1

After $n_{k - 1}$ , choose $n_{k}$ least with $P (∣ X_{n_{k}} - X ∣ > a_{k}) < b_{k}$ ; [L1] makes every choice possible.

L1construct

2.1

The proof of [L2] uses only that the displayed probabilities are summable and that $a_{k} \to 0$ . Thus it applies to these $n_{k}$ and gives almost-sure convergence.

step 1.1L2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

S. Roch, Lecture 3: Modes of convergence, Theorem 3.12 (standard reference, not scraped)