Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ two (not necessarily absolute) convergent series. We denote their Cauchy product by
$\sum_{n=0}^\infty c_n$
where $c_n=\sum_{l=0}^n a_lb_{k-l}$.
Assume that $\sum_{n=0}^\infty c_n$ converges. Is it true that in this case we have that
$(\sum_{n=0}^\infty a_n)\cdot (\sum_{n=0}^\infty b_n)=\sum_{n=0}^\infty c_n$.
Can this be seen "easily" (in an elementary way)?