Could someone give (or point me to) a proof of this?
If $\;\; \sum_{i=1}^\infty \sum_{j=1}^\infty |f(i,j)| = L < \infty$
and $\;\; k : N → N^2 \;\;$ is a bijection,
then $\;\; \sum_{i=1}^\infty|f(k(i))| = L$.
You can assume already proved the standard inversion of limits theorem for uniform convergent sequences of functions.
Just observe that $\sum _{i=1}^{N} |f(k(i))| \leq L$ for each $n$ so $\sum _{i=1}^{\infty} |f(k(i))| \leq L$ and $\sum_{i=1}^{N} \sum_{j=1}^{N} |f(i,j)| \leq \sum _{i=1}^{\infty} |f(k(i))|$ which gives the reverse inequality. No theorem is required at all for this.