CorollaryStatement: AI-adaptedProof: 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.

Every rational approximation to a real exponent gives the same limiting real power

Statement

Let $a > 0$ , let $x \in R$ , and let $(q_{n})$ be a rational sequence converging to $x$ . Then $a^{q_{n}} \to a^{x}$ . Hence the limit is independent of the rational approximating sequence and equals the rational-supremum value $a^{[x]}$ .

Facts & Assumptions

Given: $a > 0$ , $x \in R$ , and rational $q_{n} \to x$ .

[L1]

The rational-supremum and exponential constructions agree (The rational-supremum construction of real powers agrees with the exponential construction).

[L2]

Rational powers agree with real powers, and $t \mapsto a^{t}$ is continuous (The exponential definition of real powers agrees with the existing rational powers, Continuity and derivatives of positive-base real powers).

Proof

technique · direct

1.1

For every $n$ , rational-exponent agreement gives $a^{q_{n}} = \exp (q_{n} \log a)$ .

1.2

Continuity of $t \mapsto a^{t}$ gives $a^{q_{n}} \to a^{x}$ .

2.1

Since $a^{[x]} = a^{x}$ , this limit is independent of the chosen rational sequence and has the asserted supremum value.

step 1.2L1∎

Depends on

Used by

Two to the square root of two from rational suprema and from exp(sqrt(2) log 2) Example

Dependency tree · two levels

17 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

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)