I stumbled upon this succinct proof while discussing topology with some friends.
Suppose that, for any infinite collection of sets , the cartesian product, i.e. the collection of all choice functions on , is empty. I claim this implies Tychonoff’s theorem, which states that the product of any collection of compact topological spaces is a compact topological space.
To see this, we note that Tychonoff’s theorem holds easily for any finite collection of topological spaces. Now, for any infinite collection of compact topological spaces, the product of all of these is empty by assumption, hence compact.
Thus, Tychonoff’s theorem is shown to be true. This implies the axiom of choice, which contradicts our hypothesis.
By Modus Tollens, we thus conclude that there exists some infinite collection of sets, the cartesian product of all of which is nonempty.