Prove that $\mathbb{Q}^n$ is a countable dense set in $\mathbb{R}^n$

Question

Prove that $\mathbb{Q}^n$ is a countable dense set in $\mathbb{R}^n$.

The pre-requisites are $\mathbb{Q}$ is a dense subset of $\mathbb{R}$ and $\mathbb{Q}$ is countable. I need to show

1. $\mathbb{Q}^n$ is countable, and
2. $\mathbb{Q}^n$ is dense in $\mathbb{R}$.

For the second part, if $x=(x_1,x_2,\dots,x_n)\in\mathbb{R}^n$ and $\varepsilon>0$ then, I choose $q=(q_1,q_2,\dots,q_n)\in\mathbb{Q}^n$ where, $$x_i<q_i<x_i+\frac{\varepsilon}{\sqrt{n}},~~q_i\in\mathbb{Q}.$$ Then, $\|x-q\|<\varepsilon$ that is, $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$.

I am not sure how to prove that $\mathbb{Q}\times \mathbb{Q}\times \mathbb{Q}\dots_{n\mathrm{~times}}$ is countable. Induction seems to be the possible answer; I think I might be able to go through it if I could prove it for $\mathbb{Q}^2$. How do I do that?

Answer 1

Your proof for the second part looks good.

You're correct that it suffices to prove it for $\mathbb Q^2$ by induction, and in fact it suffices to show that, if $A$ and $B$ are countable, then $A\times B$ is countable. The "nicest" example of a countable set is $\mathbb N$, so you'd be done if you show that $\mathbb N\times \mathbb N$ is countable.

Can you do this using something similar to the proof that $\mathbb Q$ is countable?