LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every Cauchy sequence of rationals is bounded

Statement

Every Cauchy sequence $(a_{n})_{n \geq 1}$ of rational numbers (Cauchy sequence of rationals) is bounded: there exists a rational $M \geq 0$ such that $∣ a_{n} ∣ \leq M$ for all $n \geq 1$ .

Facts & Assumptions

Given: A Cauchy sequence $(a_{n})_{n \geq 1}$ of rational numbers.

[A1]

For every rational $ε > 0$ there exists $N \in N$ with $∣ a_{m} - a_{n} ∣ < ε$ for all $m, n \geq N$ .

[L1]

Triangle inequality on $Q$ : $∣ x ∣ \leq ∣ x - y ∣ + ∣ y ∣$ (Absolute value and the triangle inequality).

Proof

technique · direct

1.1

Apply [A1] with $ε = 1$ : fix $N \in N$ such that $∣ a_{m} - a_{n} ∣ < 1$ for all $m, n \geq N$ .

1.2

For every $n \geq N$ : $∣ a_{n} ∣ \leq ∣ a_{n} - a_{N} ∣ + ∣ a_{N} ∣$ .

2.1

For every $n \geq N$ : $∣ a_{n} ∣ < ∣ a_{N} ∣ + 1$ .

step 1.1step 1.2algebra

2.2

Define $M = \max {∣ a_{1} ∣, ∣ a_{2} ∣, \dots, ∣ a_{N - 1} ∣, ∣ a_{N} ∣ + 1}$ , a maximum of finitely many rationals, hence rational and $\geq 0$ .

step 1.1choose

3.1

For every $n < N$ : $∣ a_{n} ∣ \leq M$ , since $∣ a_{n} ∣$ appears in the maximum.

step 2.2algebra

3.2

For every $n \geq N$ : $∣ a_{n} ∣ < ∣ a_{N} ∣ + 1 \leq M$ .

step 2.1step 2.2algebra

4.1

For every $n \geq 1$ : $∣ a_{n} ∣ \leq M$ , so $(a_{n})$ is bounded.

step 3.1step 3.2∎

Depends on

Used by

Null sequences form an ideal Lemma
Cauchy sequences form a commutative ring Theorem

Dependency tree · two levels

4 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., §5.1 (standard reference, not scraped)
Cauchy sequence (Wikipedia) (standard reference, not scraped)