A Real Hilbert Space with an Orthonormal Basis is Separable

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$, and let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$. Let $\mathbb{Q}$ be the set of rational numbers, and for $n\in\mathbb{N}$ let $\mathbb{Q}^{n}$ be the set of $n$-tuples in $\mathbb{Q}$. Let $D$ be the set of those $z\in H$ for which there are $n\in\mathbb{N}$ and $q\in\mathbb{Q}^{n}$ with

$$z=\sum_{k=1}^{n}q_{k}e_{k}.$$

Then the following hold.

1. (A countable dense set)(i) The set $D$ is countable and dense in $H$.

2. (Separability)(ii) The metric space $(H,d)$ is separable.