The problem is:
Show, assuming AC, that any Cartesian product of finite sets is either finite or uncountable (can’t be countably infinite).
My thought is: Finite product of finite sets is finite. This can be readily proved. Countably infinite product of finite sets is uncountable. This can be seen from the infinite sequence of 0 and 1. Uncountable product or with more than two elements is also uncountable. Trivially any Cartesian product of a set with only one element is finite. Then I'm done.
But I didn't use the ac. There should be a formal proof with ac. How to use ac? Please give me a hint.