Let $A_j$ be a sequence in $\mathbb {C}^{n\times n}$. Show that $\displaystyle \sum_{j=0}^\infty A_j$ converges if $\displaystyle \sum_{j=0}^\infty ||A_j||$ does.
Note that $\displaystyle \Vert A\Vert=\sup_{|x|=1} \vert Ax \vert$. I know that since $\displaystyle \sum_{j=0}^\infty ||A_j||$ does converge, then $\displaystyle \Vert A_j \Vert$ eventually converges to zero. Where do I go from here?