Inequality of $|e^z - 1|$ I'm doing a problem related with Rouché Theorem and to prove it I need to show the following inequality:
$|e^z - 1| < e - 1$ if $|z| < 1$.
I've seen that $|e^z - 1| \leq e + 1$ and that $|e^z| \leq e$ if $|z| < 1$. But I can't manage to get that $|e^z - 1| < e - 1$.
I would be really grateful if someone could help me. Thanks in advance!

$|e^z-1|=|\sum_{n \ge1}\frac{z^n}{n!}| \le \sum_{n \ge1}|\frac{z^n}{n!}| < \sum_{n \ge1}\frac{1}{n!}=e-1$ since $|z| < 1$ so $|z|^n <1$ for all $n$
