Euclidean Space is a Separable Metric Space

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, 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)(i) The metric space $(\mathbb{R}^{n},d_{E})$ is separable.