TITLE: $\mathbb{Q}^2$ is dense in $\mathbb{R}^2$
QUESTION [6 upvotes]: I know that $\mathbb{Q}$ is dense in $\mathbb{R}$. What is the next step to prove that $\mathbb{Q}^2$ is also dense in $\mathbb{R}^2$?
Any hints are welcomed.

REPLY [3 votes]: Let $x \in \mathbb{R}^2$, and let $q_n, r_n \in \mathbb{Q}$ such that $q_n \to x_1, r_n \to x_2$.
Then $\|x-(q_n,r_n)\| = \sqrt{(x_1-q_n)^2+(x_2-r_n)^2} \to 0$, hence the desired result.