TheoremStatement: 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.

Limits in probability are unique almost surely

Statement

If $X_{n} \to X$ and $X_{n} \to Y$ in probability, then $X = Y$ almost surely.

Facts & Assumptions

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

[L1]

Convergence in probability means every fixed positive error probability tends to zero (Convergence in probability).

Proof

technique · direct

1.1

For $a > 0$ , the triangle inequality gives the containment [given] ${∣ X - Y ∣ > a} \subseteq {∣ X - X_{n} ∣ > a / 2} \cup {∣ X_{n} - Y ∣ > a / 2} .$

given

2.1

Taking probabilities in step 1.1 and then limits gives the following. [step 1.1, L1] $P (∣ X - Y ∣ > a) = 0$ for every $a > 0$ . The union over $a = 1 / m$ is ${X \neq Y}$ , so it is null.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Section 1.3 (standard reference, not scraped)