If $|A \times B|$ is finite, does it follow that $A$ and $B$ are finite?
This is a question from Munkres.
I was wondering if this is a reasonable argument. Without loss of generality let us first assume that $A$ and $B$ are nonempty. First note that $A=A \times \emptyset$. Then we see that $A \times \emptyset \subseteq A \times B$. Since $|A \times B|=M$ and since $A \times \emptyset$ is a subset of $A \times B$, then $|A \times \emptyset|\leq M$.
This is one of those problems which I believe to be true, but I'm not sure how to prove it rigorously.