countable sets

I need some help with this proof:

Let C be a countable set. Prove C^n is countable for all n.

I was thinking to show that the Cartesian product of any countable sets is countable by writing them out in a table and going through the elements diagonally to count them, similar to a proof I have seen to show that the rational numbers are countable.

this proves C^2 is countable since C is countable.
then using induction I can say that C^n+1 is countable because it = C^n (which is countable from the induction premise) x C and is therefore the cartesian product of two countable sets.

Quote:

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Originally Posted by stuffthings

Let C be a countable . Prove C^n is countable for all n.

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Suppose that $\displaystyle \mathbb{N}$ is the set of natural numbers.
Map $\displaystyle \mathbb{N}\times\mathbb{N}\to\mathbb{N}$ by $\displaystyle (j,k)\mapsto 2^j\cdot 3^k$.
Show that is an injection.
Thus $\displaystyle C\times C $ is countable. Now use induction.