DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Limit superior and limit inferior of a sequence of sets

Definition

For a sequence $(A_{n})_{n \in N}$ of subsets of a set $X$ , define

${lim inf}_{n \to \infty} A_{n} : = ⋃_{n \in N} ⋂_{k \geq n} A_{k}, {lim sup}_{n \to \infty} A_{n} : = ⋂_{n \in N} ⋃_{k \geq n} A_{k} .$

The sequence begins at index $0$ . If the two sets are equal, their common value is called the limit of the sequence of sets.

Used by

Dependency tree · next 3 levels

Nothing. This result depends on no other item in the library.

Sources

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