Let $\sum\limits_{n=0}^\infty a_n$ and $\sum\limits_{n=0}^\infty b_n$ be absolutely convergent series of complex numbers. For each $n \in N$, set $c_n = \sum\limits_{k=0}^n a_k b_{n-k}$. Prove that $\sum\limits_{n=0}^\infty c_n$ is absolutely convergent and that $\sum\limits_{n=0}^\infty c_n = (\sum\limits_{n=0}^\infty a_n )(\sum\limits_{n=0}^\infty b_n )$
I'm not familiar with complex analysis (although I imagine the proof here doesn't really revolve around the series being complex). I think I can understand the second part, as each term in $(a_0 + a_1 + a_2 +...)(b_0 + b_1 + b_2 +...)$ will appear exactly once in $\sum\limits_{n=0}^\infty c_n$ However, what is the easiest way of showing that the series converges in the first place? It's been a good while since I've done analysis so all help is appreciated!