ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Even and odd subsequences of $1 / (k + 1)$ converge to $0$

Example

Let $(x_{k})$ be the null sequence of The sequence $1 / (k + 1)$ converges to $0$ , let $n : N \to N$ be the even strictly increasing index map constructed in FALSE: every bounded sequence converges, and let $m_{j} : = σ (n_{j})$ be its successor, the odd index map. Then the subsequences $x_{n_{j}}$ and $x_{m_{j}}$ both converge to $0$ (Subsequences inherit the limit): removing infinitely many indices from this already convergent sequence, even all odd ones, preserves its inherited limit. A convergent subsequence of a divergent sequence can exist, so no converse is asserted, and by A strictly increasing index map satisfies $n_{k} \geq k$ the index maps satisfy $n_{j} \geq j$ and $m_{j} \geq j$ , so the subsequences are genuine thinnings of the sequence.

Facts & Assumptions

Given: The null sequence $x_{k} : = (k + 1)^{- 1}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), the index map $n$ with $n_{0} = 0$ and $n_{σ (j)} = σ (σ (n_{j}))$ constructed in FALSE: every bounded sequence converges, and the successor map $m_{j} : = σ (n_{j})$ .

[F1]

$(x_{k})$ converges to $0$ (The sequence $1 / (k + 1)$ converges to $0$ ).

[F2]

The map $n$ satisfies $n_{0} = 0$ , $n_{σ (j)} = σ (σ (n_{j}))$ for every $j$ , and $n$ is strictly increasing (FALSE: every bounded sequence converges).

[F3]

Subsequences inherit limits: if $(z_{k})$ converges to $z$ and $p : N \to N$ is strictly increasing, then $(z_{p_{j}})$ converges to $z$ (Subsequences inherit the limit).

[F4]

Growth of index maps: if $p$ is strictly increasing then $p_{j} \geq j$ for every $j$ (A strictly increasing index map satisfies $n_{k} \geq k$ ).

[F5]

Consecutive comparisons suffice for strict increase: if $p_{i} < p_{σ (i)}$ for every $i$ , then $p$ is strictly increasing (A strictly increasing index map satisfies $n_{k} \geq k$ ).

[F6]

Strictly increasing means $p_{i} < p_{j}$ whenever $i < j$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[F7]

Discreteness: for naturals $a, b$ , $a < b ⟺ σ (a) \leq b$ (Discreteness: $σ (n)$ is the immediate successor).

Verification

technique · direct

1.1givenF1F2

$(x_{k})$ converges to $0$ and $n$ is strictly increasing.

1.2givenF2F5F7

For every $j$ one has $m_{j} = σ (n_{j}) = n_{j} + 1$ , so $m$ is strictly increasing: from $n_{j} < n_{σ (j)}$ and [F7] we get $m_{j} = σ (n_{j}) \leq n_{σ (j)} < σ (n_{σ (j)}) = m_{σ (j)}$ , that is $m_{j} < m_{σ (j)}$ at every $j$ , so [F5] applies.

2.1step 1.1F3

The subsequence along $n$ converges to $0$ : [F3] applied to step 1.1 gives $x_{n_{j}} \to 0$ .

2.2step 1.1step 1.2F3

The subsequence along $m$ converges to $0$ : [F3] applied to step 1.1 and the strict increase of step 1.2 gives $x_{m_{j}} \to 0$ .

3.1step 1.1step 1.2step 2.1step 2.2F4F6∎

Both index maps grow at least like the identity: $n_{j} \geq j$ by [F4] applied to the strict increase of $n$ in step 1.1, and $m_{j} \geq j$ by [F4] applied to the strict increase of $m$ in step 1.2. Hence the even and odd subsequences of [F6] are genuine thinnings of $(x_{k})$ and both converge to the inherited limit $0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 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

John K. Hunter, An Introduction to Real Analysis (complete author-hosted lecture notes, 307 pp.) (standard reference, not scraped)
Jiri Lebl, Basic Analysis I: Introduction to Real Analysis, Volume I (author-hosted complete book PDF) (standard reference, not scraped)

Even and odd subsequences of 1/(k+1)1/(k+1) converge to 00