PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Set liminf means eventual membership, set limsup means repeated membership, and liminf is contained in limsup

Statement

For a sequence $(A_{n})_{n \in N}$ of subsets of $X$ and $x \in X$ :

$x \in {lim inf}_{n} A_{n}$ if and only if there is $N \in N$ such that $x \in A_{k}$ for every $k \geq N$ ;
$x \in {lim sup}_{n} A_{n}$ if and only if for every $N \in N$ there is $k \geq N$ with $x \in A_{k}$ , equivalently $x$ belongs to infinitely many terms;
${lim inf}_{n} A_{n} \subseteq {lim sup}_{n} A_{n}$ .

Facts & Assumptions

Given: A sequence $(A_{n})_{n \in N}$ of subsets of $X$ and a point $x \in X$ .

[L1]

The definitions are ${lim inf}_{n} A_{n} = ⋃_{n} ⋂_{k \geq n} A_{k}$ and ${lim sup}_{n} A_{n} = ⋂_{n} ⋃_{k \geq n} A_{k}$ (Limit superior and limit inferior of a sequence of sets).

Proof

technique · direct

1.1L1

By [L1], $x \in {lim inf}_{n} A_{n}$ exactly when $x$ belongs to one tail intersection, which is exactly the existence of $N$ such that $x \in A_{k}$ for every $k \geq N$ . This proves both directions of claim 1, including $N = 0$ .

1.2L1

By [L1], $x \in {lim sup}_{n} A_{n}$ exactly when $x$ belongs to every tail union, which is exactly: for every $N$ there is $k \geq N$ with $x \in A_{k}$ . This is equivalent to membership in infinitely many terms, since a finite set of successful indices has an index larger than all its members.

2.1step 1.1step 1.2∎

Eventual membership from step 1.1 implies the repeated-membership condition of step 1.2 by taking $k \geq \max {N, N_{0}}$ for each requested $N_{0}$ . Hence ${lim inf}_{n} A_{n} \subseteq {lim sup}_{n} A_{n}$ .

Depends on

Limit superior and limit inferior of a sequence of sets

Used by

Measures converge for a convergent sequence of sets contained in one finite-measure set Corollary

Dependency tree · one level

1 result within one dependency step 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.9 (standard reference, not scraped)