Proving inequalities hold when applying exponentials

Question

So I'm set out to prove that for all $a,b\in\mathbb{R}^+$ where $a,b > 0$, and for all $r\in\mathbb{Q}$ where $r > 0$,

$$ a < b \quad \text{if and only if} \quad a^r<b^r $$

This seems so obvious that it shouldn't have to be proved, and thus I'm not really sure how to start.

Answer 1

The hint: $$b^r-a^r=a^r\left(\left(\frac{b}{a}\right)^r-1\right).$$

Answer 2

A strategy can be like this:

1. prove that it holds for $r\in\Bbb N\setminus\{0\}$;

2. use (1) to prove that it holds when $\frac1r\in\Bbb N$;

3. use (1) and (2) to prove that it holds for all $r\in\Bbb Q_{>0}$.