Prove if $a>1, r,s\in \mathbb{Q}, r>s>0$ , then $a^r>a^s>1$

Question

I have proven that if $a>1$ and $m>n$, then $a^m>a^n>1$ with $m,n \in\mathbb{N}$

But I am having severe problems when I am trying to prove it for $a>1, r,s\in \mathbb{Q}, r>s>0$ , then $a^r>a^s>1$...

Any impulses, suggestions?

Answer 1

If $r=m/n>p/q=s$, then $mq>np$ with $mq,np\in\mathbb{N}$. You have proved $b^{mq}>b^{np}>1$ for $b>1$. Let $a>1$. Choose $b=a^{1/nq}>1$, what do you get?