The Cauchy criterion for double series is the following:
I am wondering how this criterion relates to the fact that for the comparison test to be applied to a double series, the terms of the series must be nonnegative? Or is there another Cauchy criterion that is used?
Suppose $|b_{kl}| \leqslant a_{kl}$ and $\sum_{k,l}a_{kl}$ converges, then for every $\epsilon > 0$ there exists a positive integer $M$ such that if $p,q > M$ we have for all $r,s \geqslant 1$
$$\left|\sum_{k=1}^{p+r}\sum_{l=1}^{q+s}b_{kl} - \sum_{k=1}^{p}\sum_{l=1}^{q}b_{kl}\right| \\= \left|\sum_{k=p+1}^{p+r}\sum_{l=q+1}^{q+s}b_{kl} + \sum_{k=1}^{p}\sum_{l=q+1}^{q+s}b_{kl} + \sum_{k=p+1}^{p+r}\sum_{l=1}^{q}b_{kl} \right| \\ \leqslant \sum_{k=p+1}^{p+r}\sum_{l=q+1}^{q+s}|b_{kl}| + \sum_{k=1}^{p}\sum_{l=q+1}^{q+s}|b_{kl}| + \sum_{k=p+1}^{p+r}\sum_{l=1}^{q}|b_{kl}|\\ \leqslant \left|\sum_{k=1}^{p+r}\sum_{l=1}^{q+s}a_{kl} - \sum_{k=1}^{p}\sum_{l=1}^{q}a_{kl}\right| < \epsilon. $$
Thus, $\sum_{k,l} b_{kl}$ converges by the Cauchy criterion.
The terms $b_{kl}$ need not be nonnegative to apply the comparison test which proves absolute convergence. Of course, the terms $a_{kl}$ used for the comparison are nonnegative.