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Let us consider the set $c$ of convergent sequences, and the subspace $c_0$ of convergent sequences to zero. They are Banach spaces over $\mathbb C$ or $\mathbb R $ under the sup-norm (and the usual vector space operations).

I want to know which of these spaces are separable. I think that $c$ is not separable but I can't prove it. Please help me!

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They're both separable. Let $\mathcal S$ be the set of all sequences which (a) consist entirely of rational numbers, and (b) are eventually constant.

It's fairly straightforward to show that this set is countable, and is dense in $c$. The subset where they're eventually constantly zero is dense in your second space.

Let me know if you need more details.

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