Is $\Bbb Q^n$ dense in $\mathbb{R}^n$ for $n>1$?

Question

It is well known that the rational numbers, $\mathbb{Q}$, are dense in $\mathbb{R}$.

My question here :Is $\ \mathbb{Q}^n$ dense in $\mathbb{R}^n$ for $n>1$ ?

Edit : I edited the question as it is related to the precedent question.

Thank you for any help .

Answer 1

That is pretty trivial. If $\mathbb{Q}$ is dense in $\mathbb{R}$, for any $(x_1,\ldots,x_n)\in\mathbb{R}^n$ we may find $(q_1,\ldots,q_n)$ such that $|x_i-q_i|\leq\varepsilon$, for any $\varepsilon>0$, so $\|x-q\|_1\leq n\varepsilon $ is arbitrarily small.

Answer 2

More generally, the product of dense sets is dense in the product topology.

For a proof, see this post.