The Cartesian product of a finite number of countable sets is countable
Solution 1:
Let $A$ and $B$ be countable, i.e. we have the enumerations
$$a_1,a_2,a_3,\cdots$$ and $$b_1,b_2,b_3,\cdots$$
Then we can enumerate $A\times B$ as
$$(a_1,b_1),(a_2,b_1),(a_1,b_2),(a_3,b_1),(a_2,b_2),(a_1,b_3),\cdots$$
(notice the sums of the indexes, $2,3,3,4,4,4,\cdots$). You can easily check that all pairs $(a_n,b_m)$ are cited.
Then by induction,
$$\prod_{i=1}^n A_i=\left(\prod_{i=1}^{n-1} A_i\right)\times A_n.$$ is countable.