ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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.

x sin(1/x) tends to zero despite its oscillation

Example

$\lim_{x \to 0} x \sin (1 / x) = 0,$ where the function is defined at $0$ by the displayed limit.

Facts & Assumptions

Given: A nonzero real $x$ approaching $0$ .

[L1]

$∣ \sin u ∣ \leq 1$ for every real $u$ (Parity and the Pythagorean identity for sine and cosine).

[L2]

The squeeze theorem for function limits holds (If $f \leq g \leq h$ near $c$ and $f$ and $h$ have the same limit at $c$ , then so does $g$ ).

Verification

technique · direct

1.1

From [L1], $∣ x \sin (1 / x) ∣ \leq ∣ x ∣$ for every $x \neq 0$ .

L1algebra

2.1

Since both $- ∣ x ∣$ and $∣ x ∣$ tend to $0$ , squeeze gives $x \sin (1 / x) \to 0$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

C. Schmeiser, Introduction to Analysis (standard reference, not scraped)
R. Bartle and D. Sherbert, Introduction to Real Analysis (standard reference, not scraped)
H. Zeisel, lim sin(x)/x and the definition of pi (standard reference, not scraped)