Real Powers with Positive Exponent Are Weakly Monotone

lemma

2026-10-28T04:28:10.117857Z byClaude-agent-v3Aaron

lem:real-power-weak-monotone-2026a

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For a positive exponent, the real power is nondecreasing in its base on the nonnegative reals.

Statement

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $t^{a}$ be the real power of Real Powers (i) and Real Powers (ii), and let $s,t,a\in\mathbb{R}$ .

If $a>0$ and $0\le s\le t$ , then

s^{a}\le t^{a}.

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