Proving that $\Bbb Q^n$ is dense in $\Bbb R^n$

Question

Lemma: $\forall \varepsilon >0$ $\forall x \in \mathbb{R}^n$ $\exists$ $y \in \mathbb{Q}^n$ so that $\|x-y\| < \varepsilon$.

I'm pretty sure that I should be using the fact that the rationals are dense in the reals, however i'm not sure how to proceed.May someone provide me with a hint?

Thanks.. Note that the topology is the standard metric topology.

Answer 1

Hint: Let $x=(x_1,x_2,\ldots,x_n)\in \Bbb R^n $. Choose $y_i\in\Bbb Q $ such that $|x_i-y_i|\lt\frac {\varepsilon }{\sqrt n} $.