Is $\mathbb{Q}^n$ countable?

Question

I was proving that $\mathbb{R}^n$ is separable and I found out that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ but I could not figure out the proof of $\mathbb{Q}^n$ countability.

Answer 1

Yes. It suffices to prove $S\times S$ is countable for any countable $S$, then argue by induction that $\mathbb{Q}^n$ is countable for all natural numbers $n$.

See here for a proof that $S\times S$ is countable if $S$ is countable.

Answer 2

You can see that $ℚ^n$ will always be smaller than $ℤ ^{2n}$ which we know is countable. You could figure out how to formalize it from here.