ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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 summable tail bound becomes an almost-sure eventual bound

Example

Let $(X_{n})_{n \in N}$ be real random variables and let $(a_{n})_{n \in N}$ be positive real numbers. If $\sum_{n = 0}^{\infty} P (∣ X_{n} ∣ > a_{n}) < + \infty,$ then with probability $1$ only finitely many indices $n$ satisfy $∣ X_{n} ∣ > a_{n}$ . Equivalently, almost surely there is $N$ such that $∣ X_{n} ∣ \leq a_{n}$ for every $n \geq N$ .

Facts & Assumptions

Given: Real random variables $X_{n}$ and positive thresholds $a_{n}$ with $\sum_{n = 0}^{\infty} P (∣ X_{n} ∣ > a_{n}) < + \infty$ .

[L1]

If the probabilities of events are summable, then the corresponding events occur infinitely often with probability $0$ . (First Borel-Cantelli lemma for events)

Verification

technique · direct

1.1

Apply [L1] to the events $A_{n} : = {∣ X_{n} ∣ > a_{n}}$ . Then $P (∣ X_{n} ∣ > a_{n} i . o .) = 0.$

2.1

The complement of the event in step 1.1 is exactly the event that there is an index $N$ such that $∣ X_{n} ∣ \leq a_{n}$ for all $n \geq N$ . Hence that eventual bound holds almost surely.

step 1.1algebra∎

Depends on

First Borel-Cantelli lemma for events

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.