My goal is to prove that $|S\times S| = |S|$ if $S$ is Dedekind-infinite. In this case Dedekind-infinite means either that $|\mathbb{N}| \leq |S|$ or equivalently that there exists a proper subset $T \subset S$ such that $|T| = |S|$.
It is enough to show that an injection from $S\times S$ to $S$ exists (the other injection from $S$ to $S \times S$ is trivial, then one can apply Cantor-Bernstein). However I have no idea how to construct such an injection except when $S$ is countable.
I'm looking for a rigorous proof, but any help is appreciated.
Thanks!
