Show that if $\displaystyle |a|<1$ and $\displaystyle |b|<1$ , then

$\displaystyle \frac{|a|-|b|}{1-|ab|} \le \frac{|a+b|}{|1+ab|} \le \frac{|a|+|b|}{1+|ab|}$.

I try to prove by squaring the terms but it is long. Do anyone has a shorter approach to the question?