DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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 nonnegative extended-real sequence

Definition

Let $(a_{n})_{n \in N}$ be a sequence in $[0, + \infty] \subseteq \overline{R}$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined). For each $N \in N$ , completeness of $\overline{R}$ (Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ ) gives the tail bounds

$u_{N} : = \sup_{k \geq N} a_{k}, ℓ_{N} : = \inf_{k \geq N} a_{k} .$

The limit superior and limit inferior of $(a_{n})$ are

${lim sup}_{n \to \infty} a_{n} : = \inf_{N \in N} u_{N}, {lim inf}_{n \to \infty} a_{n} : = \sup_{N \in N} ℓ_{N} .$

All four suprema and infima exist in $\overline{R}$ , including when some terms are $+ \infty$ . This definition is therefore distinct from the real-sequence definition: it applies directly to sequences of measure values.

Depends on

Used by

Dependency tree · two levels

12 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

G. Folland, Real Analysis, 2nd ed., §1.3, Exercise 8 (standard reference, not scraped)