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

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

Your help is much appreciated!