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I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space $(X,d_\infty)$, which consists of all sequences with a limit in $\mathbb R$ is complete. How can I prove that $c_0$ is a closed subspace of of $X$?

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The complement is open: take $x=\{x_n\}$ a sequence which doesn't converge to $0$: we can find $\delta_0>0$ and an infinite subset $A$ of the natural numbers such that $|x_n|\geq 2\delta$ for all $n\in A$. The open ball of center $x$ and radius $\delta$ contains sequences which are not converging to $0$, as we have $|y_n|\geq\delta$ for all $n\in A$ and $y=\{y_n\}\in B_{\infty}(y,\delta)$.

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Alternatively, observe that $c_0$ is the kernel of the continuous linear functional $\lim\colon X \to \mathbb{R}$. – t.b. Sep 10 '12 at 21:31
Or alternatively show that it contains all its limit points, that is, the limit of a sequence in $c_0$ is in $c_0$ again. You can also have a look at this to get ideas of how to show a set is closed. – Rudy the Reindeer Sep 11 '12 at 5:34

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