[Math] Prove that $c$ is separable

functional-analysis

I need to prove that $c$ – space of all convergent sequences – is seperable.

I believe that $c$ is a subspace of $\ell^1$.
Now, $\ell^1$ is separable, so $c$ is also separable.

Edited:

So let $R$ be a set of all sequences $r=(r_1,r_2, \dots,r_n,r_n,\dots)$ where $r_i$ are rational. $R$ is countable.

Now take $\epsilon >0$ and $x\in c$.
$\mathbb{Q}$ is dense in $\mathbb{R}$, so there exists $r \in R$ such that $\sup_n |r_n-x_n|< \epsilon$

Edit I know its wrong..

I believe that $c$ is a subspace of $\ell^1$. Now, $\ell^1$ is separable, so $c$ is also separable.

The problem is that there are sequences $(a_n)$ which converge to $0$ but so slowly that the series $\sum_n |a_n|$ is divergent.

However, we can consider the subset of sequences with rational entries, and which are eventually constant. This is a countable dense subset.