Suppose we have the following
$$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$
where all the $a_{ij}$ are non-negative.
We know that we can interchange the order of summations here. My interpretation of why this is true is that both this iterated sums are rearrangements of the same series and hence converge to the same value, or diverge to infinity (as convergence and absolute convergence are same here and all the rearrangements of an absolutely convergent series converge to the same value as the series).
Is this interpretation correct. Or can some one offer some more insightful interpretation of this result?
Please note that I am not asking for a proof but interpretations, although an insightful proof would be appreciated.
Best Answer
This isn't a proof, but perhaps can give you the insight you are looking for. Any nondecreasing sequence converges to its (possibly infinite) supremum. Thus a series of nonnegative terms converges to the supremum of its partial sums and interchanging the order of summation doesn't affect the value of the supremum: there is no accidental cancellation of terms of opposite sign.