CounterexampleConstruction: 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.

sin(1/x) has no limit as x tends to zero

Statement refuted

The function $x \mapsto \sin (1 / x)$ , defined for $x \neq 0$ , has a limit at $0$ .

Facts & Assumptions

Given: The punctured real line.

[L1]

$\sin (π / 2 + 2 m π) = 1$ and $\sin (3 π / 2 + 2 m π) = - 1$ for integers $m$ (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi).

[L2]

A function limit implies convergence along every sequence in its punctured domain approaching the point (Heine criterion: $\lim_{x \to c} f (x) = L$ iff $f (x_{k}) \to L$ for every sequence in $A ∖ {c}$ converging to $c$ ).

[L3]

Reciprocal natural-number sequences tend to zero (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Counterexample

technique · direct

1.1

For $n \in N$ , put $x_{n} = 1 / (π / 2 + 2 π (n + 1))$ and $y_{n} = 1 / (3 π / 2 + 2 π (n + 1))$ . Both sequences are nonzero and tend to $0$ .

L3algebra

1.2

Their image values are $\sin (1 / x_{n}) = 1$ and $\sin (1 / y_{n}) = - 1$ .

2.1

A common function limit at zero would force both image sequences to converge to it, which is impossible.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)