LemmaStatement: AI-adaptedProof: AI-generatedprecheck 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.

The rationals are Archimedean

Statement

For every rational $x$ there is a natural number $n$ with $x < n$ . Consequently, for every rational $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ .

Facts & Assumptions

Given: A rational $x = [(a, b)]$ with $b > 0$ .

[L1]

The order and arithmetic of $Q$ (The rationals form a totally ordered field).

[L2]

Integer facts: positive integers are exactly $ι (m)$ with $m \geq 1$ natural; nonnegative integers are the image of $N$ ; the embeddings preserve arithmetic and order (The naturals embed in the integers, The integers embed in the rationals, The integers form a totally ordered ring).

Proof

technique · direct

1.1L2

Since $b > 0$ , $b$ lies in the image of $N$ and $b \geq 1$ .

1.2L2

If $a \leq 0$ set $k = 0$ ; otherwise $a$ is a positive integer, so $a = ι (k)$ for some natural $k \geq 1$ . In both cases $a \leq k$ (as integers).

2.1step 1.1step 1.2L2

Then $a < k + 1$ and, since $b \geq 1$ and $k + 1 > 0$ , also $k + 1 \leq (k + 1) b$ .

3.1step 2.1L1

Hence $a < (k + 1) b$ , and dividing by $b > 0$ (order-scaling in the definition of the rational order), $x = [(a, b)] < [((k + 1) b, b)] = k + 1 = : n$ .

4.1step 3.1L1∎

For rational $ε > 0$ : apply the above to $x = 1 / ε$ to get $n$ with $1 / ε < n$ , hence $1 / n < ε$ .

The rationals are Archimedean

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources