Using the fact that if $\displaystyle \sum|a_n|$ converges then:

$\displaystyle |\sum_{n=0}^{\infty}a_n|\leq\sum_{n=0}^{\infty}|a_ n|$,

prove that for all $\displaystyle z\in\bar{D}(0;1)$ (punctured disc centre 0 radius 1),

$\displaystyle (3-e)|z|\leq|e^z-1|\leq(e-1)|z|$.

I haven't even got the first part yet. Ive tried manipulating e (as $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n!}$) but can't get the right answer.