LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)verified 2026-07-26 (claude-opus-5)

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.

For positive terms, null and divergence to $+\infty$ are reciprocal

Statement

Let $(x_k)$ be a sequence of reals with $x_k > 0$ for every $k \in \mathbb{N}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Order on the reals). Then

$(x_k) \text{ converges to } 0 \iff (1/x_k) \text{ diverges to } +\infty,$

with convergence as in Limits and Cauchy sequences of reals and divergence to $+\infty$ as in Divergence to $+\infty$ and to $-\infty$ .

The positivity hypothesis is essential and is not a convenience; see the remarks.

Facts & Assumptions

Given: A sequence $(x_k)$ of reals with $x_k > 0$ for every $k$ , so that each $x_k$ is nonzero and $1/x_k$ is defined (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Order on the reals, Field).

[L1]

Convergence, quantified over rational $\varepsilon > 0$ (Limits and Cauchy sequences of reals); divergence to $+\infty$ , quantified over real $M$ (Divergence to $+\infty$ and to $-\infty$ ).

[L2]

Inverses and order: $a > 0$ implies $a^{-1} > 0$ , and $0 < a < b$ implies $0 < b^{-1} < a^{-1}$ (Inverses of positives are positive, and reciprocation reverses order).

[L3]

The involution $(u^{-1})^{-1} = u$ for $u \ne 0$ , from uniqueness of multiplicative inverses (Field).

[L4]

Absolute value: $|u| = u$ when $u \ge 0$ , and $|u - 0| = |u|$ (Basic properties of the absolute value, Order on the reals).

[L5]

Small rationals: for every real $\eta > 0$ there is a rational $\varepsilon$ with $0 < \varepsilon < \eta$ , by density (The rationals embed densely in the reals) or by the Archimedean property (Every complete ordered field is Archimedean) applied to $1/\eta$ .

[L6]

Order arithmetic in $\mathbb{R}$ : trichotomy and transitivity, and $u > 0 \ge M$ gives $u > M$ (Complete ordered field (least-upper-bound property), Ordered field).

Proof

technique · direct

1.1

Since $x_k > 0$ we have $|x_k| = x_k$ and, by [L2], $1/x_k > 0$ , for every $k$ .

L2L4

2.1

Forward direction. Assume $(x_k)$ converges to $0$ and let $M \in \mathbb{R}$ be arbitrary. If $M \le 0$ , then $1/x_k > 0 \ge M$ for every $k$ by step 1.1, so the threshold $K = 0$ works. If $M > 0$ , then $1/M > 0$ by [L2]; by [L5] choose a rational $\varepsilon$ with $0 < \varepsilon < 1/M$ , and by [L1] take $K$ with $|x_k - 0| < \varepsilon$ for all $k \ge K$ ; for such $k$ , step 1.1 gives $0 < x_k = |x_k| < \varepsilon < 1/M$ , and applying [L2] to $0 < x_k < 1/M$ gives $1/x_k > (1/M)^{-1} = M$ by [L3]. In both cases there is $K$ with $1/x_k > M$ for all $k \ge K$ , so $(1/x_k)$ diverges to $+\infty$ .

step 1.1assume-hypL1L2L3L5L6

2.2

Backward direction. Assume $(1/x_k)$ diverges to $+\infty$ and let $\varepsilon > 0$ be rational. Then $1/\varepsilon > 0$ by [L2], and by [L1] there is $K$ with $1/x_k > 1/\varepsilon$ for all $k \ge K$ . For such $k$ , both $1/\varepsilon$ and $1/x_k$ are positive by step 1.1, so applying [L2] to $0 < 1/\varepsilon < 1/x_k$ gives $0 < (1/x_k)^{-1} < (1/\varepsilon)^{-1}$ , that is $0 < x_k < \varepsilon$ by [L3]; hence $|x_k - 0| = x_k < \varepsilon$ . So $(x_k)$ converges to $0$ .

step 1.1assume-hypL1L2L3L4

3.1

The two implications together give the stated equivalence.

step 2.1step 2.2∎

Remarks

Positivity is essential. Let $(x_k)$ be as in the lemma, so that $x_k > 0$ for every $k$ and $(x_k)$ converges to $0$ ; such sequences exist, $x_k = 1/(k+1)$ being the standard one (FALSE: limits preserve strict inequalities). Put $w_k := -x_k$ . Then $|w_k - 0| = |-x_k| = |x_k| = |x_k - 0|$ for every $k$ (Basic properties of the absolute value), so $(w_k)$ converges to $0$ as well, and every $w_k$ is nonzero. Yet $(1/w_k)$ does not diverge to $+\infty$ : $1/w_k = -(1/x_k)$ by field arithmetic (Field), and $1/x_k > 0$ (Inverses of positives are positive, and reciprocation reverses order), so $1/w_k < 0$ at every index, its negative $1/x_k$ being positive (Ordered field), and no threshold works even for $M = 0$ . Dropping positivity therefore breaks the forward implication outright. What survives without a sign hypothesis is the statement about absolute values: for a sequence of nonzero terms, $(|x_k|)$ converges to $0$ if and only if $(1/|x_k|)$ diverges to $+\infty$ , which is this lemma applied to $(|x_k|)$ , whose terms are positive (Basic properties of the absolute value).
The hypothesis $x_k > 0$ is imposed at every index so that $1/x_k$ is defined at every index. It is tempting to relax it to "eventually positive" by passing to a tail, and on the convergence side that is exactly Convergence depends only on the tail; but the equivalence also has a divergence side, and the corresponding tail statement for divergence to $+\infty$ (Divergence to $+\infty$ and to $-\infty$ ) is proved nowhere in this library, Convergence depends only on the tail covering convergence and the Cauchy condition only. The relaxed form is therefore not asserted here.
Taking $x_k := 1/(k+1)$ , which is null (FALSE: limits preserve strict inequalities), the lemma turns that one fact into $k + 1 \to +\infty$ . The two are the same statement seen twice, which is why this lemma is the standard bridge between the Archimedean property (Every complete ordered field is Archimedean) and statements about growth.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 28 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)
Limit of a 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)

For positive terms, null and divergence to +∞+\infty+∞ are reciprocal