I'm reading the book: "Deimling Nonlinear Analysis" and I'm blocked on a proof, at page 27, when proving Jordan's separation theorem.
At some point, we want to show that two sets, let's say $I_1$ and $I_2$, have the same cardinality, by looking at the sums indexed by the elements of these two sets.If both sets have the finite cardinality the proof is pretty straightforward.
We know in fact that $a_{ij} \in \mathbb{Z}$ and $$ \begin{cases} 1 = \sum_{j\in I_1} a_{ij} \ \text{ for every } i \in I_2\\1 = \sum_{i\in I_2} a_{ij} \ \text{ for every } j \in I_1\\\end{cases} $$
so, putting together the two equations$$\#I_2 = \sum_{i\in I_2} \sum_{j\in I_1} a_{ij} = \sum_{i\in I_1}\sum_{j\in I_2} a_{ij} = \#I_1 $$
the book now states that this implies the two sets have the same cardinality... even in the infinite case! And this gives me a lot of doubts because in this situation I have to deal with a series that diverges to infinity.
I tried to prove assume, by contradiction, that one set has finite cardinality and the other not, but even if one summation is finite, I have to rearrange infinite terms to get anything useful, which isn't justified I guess. As follows:If $I_2 = \mathbb{N}$ and $I_1= \{1,...,N_0\}$$$1 = \sum_{j=1}^{N_0} a_{ij}\Rightarrow\\\Rightarrow \lim_{k\to\infty}\sum_{i=1}^k \sum_{j=1}^{N_0} a_{ij} = \lim_{k\to\infty}k\\\Rightarrow \lim_{k\to\infty}\sum_{j=1}^{N_0}\sum_{i=1}^k a_{ij} = \infty\\\Rightarrow \sum_{j=1}^{N_0}\lim_{k\to\infty}\sum_{i=1}^k a_{ij} = \infty\\\Rightarrow\sum_{j=1}^{N_0} \sum_{i=1}^\infty a_{ij} = \infty\\\\\Rightarrow \sum_{j=1}^{N_0} 1 = \infty\\\Rightarrow N_0 = \infty$$And this should be a contradiction. BUT this proof is sketchy at best.
Below you can find a screenshot of the proof.Can anyone please help me?
