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

Your help is much appreciated!