LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Weighted ell-2 decay implies absolute convergence

Statement

Let $s > 1 / 2$ . If a sequence $(a_{k})$ satisfies $\sum_{k} (1 + ∣ k ∣)^{2 s} ∣ a_{k} ∣^{2} < \infty$ , then $(a_{k}) \in ℓ^{1} (Z)$ .

Facts & Assumptions

Given: $s > 1 / 2$ and the displayed finite weighted square sum.

Proof

technique · direct

1.1

The series $\sum_{k} (1 + ∣ k ∣)^{- 2 s}$ converges, by comparison with the integral of $x^{- 2 s}$ on $[1, \infty)$ .

givenalgebra

2.1

Cauchy--Schwarz yields $\sum_{k} ∣ a_{k} ∣ \leq {(\sum_{k} (1 + ∣ k ∣)^{2 s} ∣ a_{k} ∣^{2})}^{1 / 2} {(\sum_{k} (1 + ∣ k ∣)^{- 2 s})}^{1 / 2} < \infty .$

step 1.1algebra∎

Used by

Dependency tree · 0 levels

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

Sources

Michael E. Taylor, Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE, Section 1 (standard reference, not scraped)