TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-01

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.

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The exponential dominates every fixed nonnegative integer power at $+ \infty$

For every $m \in N$ and every real $a > 0$ , $\frac{x^{m}}{\exp (a x)} ⟶ 0 (x \to + \infty) .$

Given: $m \in N$ and $a > 0$ .

[L1]

Every term of the exponential series is nonnegative at a nonnegative argument (The real exponential function and the number $e$ by a power series).

[L2]

technique · direct

1.1L1given

For $x > 0$ , retain term $m + 1$ of the series at $a x$ : $\exp (a x) \geq (a x)^{m + 1} / ι ((m + 1)!)$ .

2.1step 1.1L2algebra

Hence $0 \leq x^{m} / \exp (a x) \leq ι ((m + 1)!) / (a^{m + 1} x)$ .

3.1step 2.1L2∎

The upper bound tends to $0$ , so the quotient tends to $0$ .

Depends on