This post is related to the question What is an easy way to prove the equality $r > s > 0$ implies $x^r > x^s$? which is essentially the same question reduced to the unit interval. In particular I was wondering if there was a simple proof for the following inequality:
Let $x \in (0,1)$ and $r,s \in \mathbb{R}$
How do you show the inequality $r > s > 0$ implies $x^r < x^s$?