LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Finite-range sequences are uniformly dense in ell-infinity

Statement

The finite-range real, respectively complex, sequences are dense in $ℓ^{\infty}$ for the supremum norm.

Facts & Assumptions

[L1]

$ℓ^{\infty}$ consists of bounded scalar sequences with the supremum norm (The sequence spaces c_0 and ell-infinity).

Proof

technique · direct

Given: The objects and hypotheses in the Statement.

1.1givenL1

Let $x \in ℓ^{\infty}$ and $ε > 0$ . In the real case partition the bounded interval containing all $x_{n}$ into finitely many half-open intervals of length below $ε$ , and replace every $x_{n}$ by a fixed endpoint of its cell. The resulting sequence $s$ has finite range and $∥ x - s ∥_{\infty} < ε$ . [L1, finite partition] (finite partition).

2.1givenL1step 1.1∎

In the complex case partition a square containing all $x_{n}$ into finitely many squares of side below $ε / \sqrt{2}$ and replace by one corner of the containing cell. Again $s$ has finite range and $∥ x - s ∥_{\infty} < ε$ . This proves density in both scalar fields. [L1, step 1.1, Euclidean estimate] (Euclidean estimate).

Depends on

The sequence spaces c_0 and ell-infinity

Used by

The finitely additive integral on ell-infinity Definition

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.

Sources

Michael Müger, Introduction to Functional Analysis (standard reference, not scraped)