# How to prove the following defined metric space is separable

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1. Oct 1, 2015

### L.S.H

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Let $\mathbb{X}$ be the set of all sequences in $\mathbb{R}$ that converge to $0$. For any sequences $\{x_n\},\{y_n\}\in\mathbb{X}$, define the metric $d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}$. Show the metric space $(\mathbb{X},d)$ is separable. I understand that I perhaps need to find a countable dense set in $\mathbb{X}$.

2. Oct 1, 2015

### Krylov

Depending on whether or not this is an exercise question, you may be able to give a quick answer by noting that the dual space of $X$ is $\ell_1$.

3. Oct 1, 2015

### Dick

Yes, that's exactly what you should do. Any thoughts? To get started do you know a countable dense subset of $\mathbb{R}$?