Why is Q dense in R? Consider a topological space $S$ and an arbitrary subset $E$. Then if the closure of $E$ given by $\bar{E} = S$, we say that $E$ is dense in $S$.
How can we prove that $\mathbb{Q}^{n}$ is dense in the vector space $\mathbb{R}^{n}$ with the standard metric topology?
I intended to prove that $\mathbb{Q}^{n}$ is dense in the vector space $\mathbb{R}^{n}$ in the topological sense, not the usual "between two $\mathbb{R}$ there is a $\mathbb{Q}$". Of course, both might be equivalent, but that remains to be proved.

You can construct $\mathbf{R}$ is many ways, see this for instance. The first one (by Cauchy sequences) is the usual one. In it (I encourage you to find it somewhere and to sudy it, as it is very enlightening), $\mathbf{Q}$ is dense in $\mathbf{R}$ by contruction. It is also the case in the second construction, from Dedekind cuts.
