How to prove this theorem using order axioms?

Question

I need to prove that if $\space\space0<a<b\space$ then $\space0<b^{-1}<a^{-1}$.

Answer 1

$ab\gt0$, so:

$$\frac{a}{ab}\lt\frac{b}{ab}$$

$$\frac{1}{b}\lt\frac{1}{a}$$