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Cartesian product proof

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I need help constructing a proof for the
Cartesian product of finitely many countable sets is countable.

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Solution Summary

This is a proof regarding the Cartesian product of countable sets.

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First let's state the counting theorem:
"The Counting Theorem":
Let S be a set. The following are equivalent.
(i) S is either finite or a countably infinite set.
(ii) Either S = Null or there is a surjective map alpha: N ---> S.
(iii) There is an injective map beta: S ---> N.
If any of these three equivalent conditions hold, then we say
that S is a countable set.
This theorem has a ...

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