LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)audited 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.

Null sequences are Cauchy

Statement

Every null sequence (Null sequence) is a Cauchy sequence (Cauchy sequence of rationals).

Facts & Assumptions

Given: A null sequence $(a_n)$ and a rational $\varepsilon > 0$ .

[A1]

Null: for every rational $\delta > 0$ there is $N$ with $|a_n| < \delta$ for all $n \ge N$ .

[A2]

Cauchy: for every rational $\delta > 0$ there is $N$ with $|a_m - a_n| < \delta$ for all $m,n \ge N$ .

[L1]

Field arithmetic in $\mathbb{Q}$ : $\varepsilon/2 > 0$ and $\varepsilon/2 + \varepsilon/2 = \varepsilon$ (The rationals form a field, The rationals form a totally ordered field).

[L2]

Triangle inequality (Absolute value and the triangle inequality).

Proof

technique · direct

1.1

$\varepsilon/2$ is a positive rational.

2.1

Fix $N$ with $|a_n| < \varepsilon/2$ for all $n \ge N$ .

step 1.1A1

3.1

For $m, n \ge N$ : $|a_m - a_n| \le |a_m| + |a_n| < \varepsilon/2 + \varepsilon/2 = \varepsilon$ .

step 2.1L1L2

4.1

$(a_n)$ is Cauchy.

step 3.1A2∎

Depends on

Used by

Null sequences form an ideal Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. Tao, Analysis I, 3rd ed., §5.2 (standard reference, not scraped)