The Real Line and Euclidean Space Are Separable

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $k\in\mathbb{N}$, and let $d_{E}$ be the Euclidean distance on $\mathbb{R}^{k}$.

(i) The real line. The real line $(\mathbb{R},d_{\mathbb{R}})$ is a separable metric space.

(ii) Euclidean space. $(\mathbb{R}^{k},d_{E})$ is a separable metric space.