convergence of double series

Prove if $\sum_{i=1}^\infty \sum_{j=1}^\infty \left| a_{ij} \right|$ converges (meaning for each fixed $i \in N$, $\sum_{j=1}^\infty \left| a_{ij} \right|$ converges to some real number $b_i$ and $\sum_{i=1}^\infty b_i$ converges as well), then $\sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij}$ and $\sum_{j=1}^\infty \sum_{i=1}^\infty a_{ij}$ both converge and $\lim_{n\to\infty} s_{nn} = \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij} = \sum_{j=1}^\infty \sum_{i=1}^\infty a_{ij}$ where $s_{nn} = \sum_{i=1}^n \sum_{j=1}^n a_{ij}$

I've been able to go along and prove that $\sum_{i=1}^\infty \sum_{j=1}^\infty \left| a_{ij} \right|$ converges implies $\sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij}$ converges, $\lim_{n\to\infty} s_{nn}$ converges, and $\lim_{n\to\infty} s_{nn} = \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij}$. However I am stuck in showing for each fixed j, $\sum_{i=1}^\infty a_{ij}$ converges to some $c_i$