LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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.

Exponential decay dominates every inverse power near zero

For every $m \in N$ , one has $β (t) / t^{m} \to 0$ as $t \to 0^{+}$ , where $β$ is the standard flat function.

Given: $m \in N$ .

[F1]

For $t > 0$ , the standard flat function satisfies $β (t) = \exp (- 1 / t)$ (The standard flat function).

[L1]

For every $m \in N$ and every $a > 0$ , $x^{m} / \exp (a x) \to 0$ as $x \to + \infty$ (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

technique · direct

1.1F1givenconstruct

For $t > 0$ , put $x : = 1 / t$ ; then $x \to + \infty$ as $t \to 0^{+}$ and $β (t) / t^{m} = x^{m} / \exp (x)$ by [F1].

2.1L1step 1.1

The expression $x^{m} / \exp (x)$ tends to $0$ by [L1] with $a = 1$ .

3.1step 2.1∎

Therefore $β (t) / t^{m} \to 0$ as $t \to 0^{+}$ .

Depends on