LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Pairing preserves convergence in probability

Statement

If $X_{n} \to X$ and $Y_{n} \to Y$ in probability, then, for every $ε > 0$ , $P ⁣ (\max {∣ X_{n} - X ∣, ∣ Y_{n} - Y ∣} > ε) ⟶ 0.$ Thus the pairs converge in probability for the max metric on $R^{2}$ .

Facts & Assumptions

Given: $X_{n} \to X$ and $Y_{n} \to Y$ in probability.

[L1]

Each coordinate convergence controls its fixed-threshold error event (Convergence in probability).

Proof

technique · direct

1.1

For $ε > 0$ , the max-metric bad event equals ${∣ X_{n} - X ∣ > ε} \cup {∣ Y_{n} - Y ∣ > ε} .$

algebra

2.1

The union bound and [L1] make the probability in step 1.1 tend to zero, which proves the claim.

step 1.1L1∎

Depends on

Convergence in probability

Used by

Continuous maps preserve convergence in probability Theorem

Dependency tree · two levels

4 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 3.14 (standard reference, not scraped)