LemmaStatement: AI-adaptedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passverified 2026-08-05 (gpt-5.6-sol-codex-subscription)

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.

Every convergent sequence is Cauchy

Statement

Let $(x_k)$ be a sequence of reals converging to $x \in \mathbb{R}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals). Then $(x_k)$ is Cauchy (Limits and Cauchy sequences of reals).

Facts & Assumptions

Given: A sequence $(x_k)$ of reals converging to a real $x$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals).

[L1]

$(x_k)$ converges to $x$ when for every rational $\varepsilon > 0$ there is $K \in \mathbb{N}$ with $|x_k - x| < \hat\varepsilon$ for all $k \ge K$ ; and $(x_k)$ is Cauchy when for every rational $\varepsilon > 0$ there is $K \in \mathbb{N}$ with $|x_k - x_l| < \hat\varepsilon$ for all $k, l \ge K$ (Limits and Cauchy sequences of reals).

[L2]

Triangle inequality: $|a + b| \le |a| + |b|$ in $\mathbb{R}$ (The triangle inequality, Complete ordered field (least-upper-bound property)).

[L3]

Absolute value: $|-u| = |u|$ for every real $u$ (Basic properties of the absolute value).

[L4]

Halving a positive rational: if $\varepsilon$ is a rational $> 0$ then $\varepsilon/2$ is again a rational, it is $> 0$ , and $\varepsilon/2 + \varepsilon/2 = \varepsilon$ . In detail, $\mathbb{Q}$ is an ordered field (The rationals form a totally ordered field, Ordered field, Field), so $1 > 0$ (The multiplicative identity is positive) and $2 := 1 + 1 > 0$ because the positives are closed under addition (Ordered field); hence $2 \ne 0$ is invertible with $2^{-1} > 0$ (Inverses of positives are positive, and reciprocation reverses order), the product $\varepsilon/2 = \varepsilon \cdot 2^{-1}$ of two positives is positive (Sign rules for products and monotonicity of multiplication), and $\varepsilon \cdot 2^{-1} + \varepsilon \cdot 2^{-1} = \varepsilon \cdot 2^{-1} \cdot 2 = \varepsilon$ by the field axioms (Field). The embedding of $\mathbb{Q}$ in $\mathbb{R}$ preserves the order (The rationals embed densely in the reals), so these facts hold verbatim for the images, under the identification recorded in Sequences of reals: bounded, eventually, frequently, tails, subsequences.

[L5]

The order on $\mathbb{N}$ is total and transitive, so a single threshold serves for both indices ( $\le$ is a linear order on $\mathbb{N}$ ).

[L6]

Order arithmetic in $\mathbb{R}$ : adding two strict inequalities, $a < b$ and $c < d$ give $a + c < b + d$ (Order is preserved by adding a constant and by adding inequalities); and, since $u \le v$ means $u < v$ or $u = v$ , the mixed form $u \le v < w \Rightarrow u < w$ (Complete ordered field (least-upper-bound property), Ordered field).

Proof

technique · direct

1.1

Let $\varepsilon > 0$ be rational; then $\varepsilon/2$ is a rational $> 0$ .

2.1

By convergence there is $K \in \mathbb{N}$ with $|x_k - x| < \varepsilon/2$ for all $k \ge K$ .

step 1.1L1

3.1

For all $k, l \ge K$ we get $|x_k - x_l| = |(x_k - x) + (x - x_l)| \le |x_k - x| + |x - x_l| = |x_k - x| + |x_l - x|$ , while adding the two strict inequalities of step 2.1 gives $|x_k - x| + |x_l - x| < \varepsilon/2 + \varepsilon/2 = \varepsilon$ ; composing the non-strict inequality with the strict one yields $|x_k - x_l| < \varepsilon$ .

step 2.1L2L3L4L5L6

4.1

Since the rational $\varepsilon > 0$ was arbitrary and the single threshold $K$ works for both indices, $(x_k)$ is Cauchy.

step 3.1L1∎

Remarks

The converse is a genuine theorem and is not proved here. "Every Cauchy sequence of reals converges" is the completeness of $\mathbb{R}$ in the Cauchy sense. It is the subject of the next page of this track, Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness, where it is proved from the least-upper-bound property, last of the four completeness results there, by way of Bolzano-Weierstrass, which is itself routed through the monotone convergence theorem. That proof is not available at this point in the reading order; the converse itself, for the $\mathbb{R}$ this library constructs, already is, by the different route the next remark records.
The converse is nevertheless already available in this library, and it should be said plainly rather than left to the next page. The $\mathbb{R}$ used throughout (The real numbers, Sequences of reals: bounded, eventually, frequently, tails, subsequences) is the quotient of the ring of Cauchy sequences of rationals, and The reals are complete proves for precisely that $\mathbb{R}$ that every Cauchy sequence of reals converges to a real. Nothing further is needed to have the converse in hand here; and any other complete ordered field inherits it, since any two are isomorphic by a unique ordered-field isomorphism (Uniqueness of the complete ordered field: $\mathbb{R}$ up to a unique isomorphism). The reason the next page proves it again, from the least-upper-bound property, is that that proof is the form the rest of analysis uses and does not route through a particular construction.
The splitting $\varepsilon/2 + \varepsilon/2$ is the whole content. It is worth noticing that no property of $\mathbb{R}$ beyond the ordered-field axioms and the triangle inequality is used, so the same argument shows that a convergent sequence of rationals is Cauchy in $\mathbb{Q}$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 29 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

J. K. Hunter, An Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)
Cauchy sequence (Wikipedia) (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §6.1 (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)