Exercise 10.N of The elements of integration and Lebesgue measure Bartle's book If $a_{mn}\ge 0$ for $m,n\in\mathbb{N}$, then $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}(\le +\infty).$
 A: Since $a_{mn}\ge 0$ for $m,n\in\mathbb{N}$, we have, for any $r,s\in\mathbb{N}$
$$\sum_{m=1}^{r} \sum_{n=1}^{s}a_{mn}= \sum_{n=1}^{s}\sum_{m=1}^{r}a_{mn} \leqslant \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}$$
and $b_{rs}=\sum_{m=1}^{r} \sum_{n=1}^{s}a_{mn}$ is a non-decreasing sequence in $r$ and in $s$. So we have, for any $r$ fixed, 
$$\sum_{m=1}^{r} \sum_{n=1}^{\infty}a_{mn}=\sum_{m=1}^{r} \left(\lim_{s\to \infty}\sum_{n=1}^{s}a_{mn} \right )=\lim_{s\to \infty}\sum_{m=1}^{r} \sum_{n=1}^{s}a_{mn} \leqslant \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}$$ 
Note now that $c_{r}=\sum_{m=1}^{r} \sum_{n=1}^{\infty}a_{mn}$ is a non-decreasing sequence in $r$, so we have 
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}=\lim_{r\to \infty}\sum_{m=1}^{r} \sum_{n=1}^{\infty}a_{mn} \leqslant \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}$$
So we have proved 
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}\leqslant \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}$$
In complete similar way, we can prove that
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}\leqslant \sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}$$
So we have 
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn} = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}$$
