Theorem:$$(\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} |a_{ij}| \space \text{converges}) \implies (\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{ij} \space \text{converges})$$
Proof:
Let $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} |a_{ij}|=L$. Then, for $s_{nn}=\sum_{i=1}^{n} \sum_{j=1}^{n} |a_{ij}|$, $\lim{s_{nn}}=L$. By the Cauchy criterion, we have that for every $\epsilon>0$, there exists an $N$ so that if $m, n > N$ and $m > n$, $|s_{mm}-s_{nn}|<\epsilon$. Thus $$|a_{(n+1)(n+1)}|+...+|a_{(n+1)m}|+...+|a_{m(n+1)}|+...+|a_{mm}|<\epsilon.$$
By the triangle inequality, $$|a_{(n+1)(n+1)}+...+a_{(n+1)m}+...+a_{m(n+1)}+...+a_{mm}|<\epsilon$$
and convergence is guaranteed for $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{ij}$.
P.S. I am slightly suspicious of this proof (it seems to only prove that a subsequence of the partial sums of $a_{ij}$ converges), but I'm not sure.