ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

Convergence of an independent random series is a zero-one event

Example

If $(X_{n})_{n \in N}$ is an independent sequence of real random variables, then the event ${\sum_{n = 0}^{\infty} X_{n} converges}$ has probability either $0$ or $1$ .

The zero-one law does not decide which value occurs; it says only that no intermediate probability is possible.

Facts & Assumptions

Given: An independent sequence of real random variables $(X_{n})_{n \in N}$ .

[L1]

Almost-sure convergence of an independent series is a tail event of probability $0$ or $1$ . (Almost-sure convergence of an independent series is a zero-one event)

Verification

technique · direct

1.1

The displayed event is exactly the convergence event covered by [L1].

2.1

Therefore its probability belongs to ${0, 1}$ .

step 1.1algebra∎

Depends on

Almost-sure convergence of an independent series is a zero-one event

Used by

Nothing in the library uses this result yet.

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.