Let $\{a_{jk}\}_{j,k \in Z_+}$ be a double sequence of complex numbers. Prove that if $\sum_{j=0}^{\infty}\sum_{k=0}^{\infty} \vert a_{jk}\vert \lt \infty$, then $\sum_{j=0}^{ \infty}\sum_{k=0}^{\infty} a_{jk} = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty} a_{jk}$.
I thought of something more or less like this
Assuming $\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{jk} = \sum_{i=0}^{\infty}a \phi(i)$ (*)
first lets show that $\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{jk} = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}a_{kj}$
só we writing $b_{jk}:= a_{kj}$, and $\sigma(i) = jk$ if $\phi(i) = kj$, we have
$b(\sigma(i)) = a(\phi*(k))$, and $\sigma$ is a one-to-one mapping of $N$ onto $N \times N$. Thus applying part (*) to b_jk and $\sigma$ we have $\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}b_{jk}=\sum_{i=0}^{\infty}b(\phi(i))$.
Is my idea correct?
Grateful for the attention.
This result is a consequence of Fubini's theorem.
What you are writing does not make a lot of sense. The line $\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{jk} = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}a_{kj}$ is merely a change of notation by switching $j$ and $k$. You cannot finish the problem by defining a bijection because series is calculated as limit of partial sum, which, a priori, can change if you just rearrange the items. You need to proceed with some different ideas.