This question already has an answer here:
Let $S=A_1 \times A_2 \times ...\times A_n$
Is S countable? And how do I prove it? I think the answer is yes because $A_1 \times A_2$ creates an infinite table so for n sets we would have an infinite table - just much bigger?
This is not a duplicate because no other question is for infinitely countable sets only