We are given the following inequality:
$|e^{x}-1|\leq|x|e^{|x|}$ where $x \in \mathbb C$
Prove that if $\sum_{n\in \mathbb N} a_{n} $ converges absolutely in $\mathbb C$ then:
$\sum_{n\in \mathbb N} (e^{a_{n}}-1) $ converges absolutely.
My thoughts: Seeing as though we have to use the above inequality it eventually boils down to showing that $|a_{n}|e^{|a_{n}|}< c_{n} $ whereby $\sum_{n\in \mathbb N} c_{n}$ converges absolutely.
I am assumung that since the series of $a_{n}$ converges absolutely then $|a_{n} | \to 0, n \to \infty$ thus implying that $|a_{n}|$ is bounded. I can't seem to find a bound that helps me along this path.
Am I on the right path, or should I take a whole new direction altogether.