LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25

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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.
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The Cauchy-sequence reals are Archimedean

Statement

The Cauchy-sequence reals $R_{C}$ (The reals form a totally ordered field) are Archimedean (Archimedean ordered field): for every $x \in R_{C}$ there is a natural number $n$ with $x < n \cdot 1$ , where the canonical natural $n \cdot 1$ is the class $\hat{n}$ of the constant rational sequence $n$ . Equivalently, the canonical naturals $(\hat{n})_{n \geq 1}$ are cofinal.

Facts & Assumptions

Given: A real $x \in R_{C}$ .

[L1]

Rational approximation: for any real $z$ and rational $ε > 0$ there is $q \in Q$ with $∣ z - \hat{q} ∣ < \hat{ε}$ , and the embedding $q \mapsto \hat{q}$ preserves and reflects order and arithmetic (The rationals embed densely in the reals).

[L2]

The rationals are Archimedean: for every rational $y$ there is a natural $n$ with $y < n$ (The rationals are Archimedean).

[L3]

$R_{C}$ is a totally ordered field, and $\hat{q} + \hat{r} = \hat{q + r}$ , $\hat{1} = 1$ (The reals form a totally ordered field, Order on the reals).

[L4]

$R_{C}$ is Archimedean iff for every real there is a natural $n$ with the real below the canonical natural $n \cdot 1 = \hat{n}$ (Archimedean ordered field).

Proof

technique · direct

1.1

By [L1] with $ε = 1$ choose a rational $q$ with $∣ x - \hat{q} ∣ < \hat{1}$ .

L1choose

1.2

By [L2] applied to the rational $q + 1$ choose a natural $n$ with $q + 1 < n$ .

L2choose

2.1

From step 1.1, $x - \hat{q} < \hat{1}$ , so $x < \hat{q} + \hat{1} = \hat{q + 1}$ .

step 1.1L3

2.2

From step 1.2, since the embedding preserves order, $\hat{q + 1} < \hat{n} = n \cdot 1$ .

step 1.2L1L3L4

3.1

Combining, $x < \hat{q + 1} < n \cdot 1$ , so $x < n \cdot 1$ for this canonical natural.

step 2.1step 2.2L3

4.1

As $x \in R_{C}$ was arbitrary, every real lies below some canonical natural: $R_{C}$ is Archimedean.

step 3.1L4∎

Depends on

Used by

The Cauchy-sequence reals have the least-upper-bound property Corollary

Dependency tree · two levels

18 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

T. Tao, Analysis I, 3rd ed., §5.4 (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
California State University San Marcos notes: Construction of the real numbers (standard reference, not scraped)