Fubini's theorem for infinite series

The extension of Fubini's theorem to infinite series is not a straightforward process. The proof acts as a summary of concepts including infinite series, series, limits, supremum and functions.

Let $f:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ be a function such that $\sum _{(n,m)\in \mathbb{N}\times \mathbb{N}}f(n,m)$ is absolutely convergent. Then we have

In other words, we can switch the order of infinite sums provided that the entire sum is absolutely convergent.