DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-24

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.

Limits and Cauchy sequences of reals

Definition

A sequence $(x_{k})$ of reals converges to $x \in R$ when for every rational $ε > 0$ there is $K$ with $∣ x_{k} - x ∣ < \hat{ε}$ for all $k \geq K$ . It is Cauchy when for every rational $ε > 0$ there is $K$ with $∣ x_{k} - x_{l} ∣ < \hat{ε}$ for all $k, l \geq K$ .

Remarks

Quantifying over rational $ε$ loses nothing: below any real $ε > 0$ lies a positive rational (The rationals embed densely in the reals).
$∣ \cdot ∣$ is the absolute value of Order on the reals.

Limits and Cauchy sequences of reals

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources