I am trying to highlight to my friend that the change of order of summation/integrals should be done with care. In that regard, the conversation moved towards the following question.
An example of a double summation with $f(m,n) > 0$ of the form $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} f(m,n)$ which diverges but $\sum_{m=1}^{\infty} f(m,n)$ converges for all $n$, and $\sum_{n=1}^{\infty} f(m,n)$ converges for all $m$.
I am not able to construct an example immediately of my head.