Is the completion of a separable normed linear space is also separable?

Question

Let $X$ be a separable normed linear space. Is the completion of $X$ is a separable Banach space?

Answer 1

There exists a countable set $B$ for $X$ : For any $x\in X$ and $\epsilon>0$, there exists $b\in B$ s.t. $$ \parallel b-x\parallel < \epsilon $$

If $\overline{X}$ is a completion and if $\overline{x}\in \overline{X}$, there exists $b\in B$ and $x\in X$ s.t. $$ \parallel b-\overline{x}\parallel \leq \parallel b-x\parallel +\parallel x-\overline{x} \parallel\leq 2\epsilon $$