Euclidean Space is a Separable Metric Space

We work in the setting of Euclidean Space and Lebesgue Measure: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the natural numbers, the real numbers with their order, and countability of sets and the Euclidean distance $d_{E}$, the metric space $(\mathbb{R}^{n},d_{E})$ and its topology are as fixed there.

1. (Separability)¶ The metric space $(\mathbb{R}^{n},d_{E})$ is separable.