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 $X$?
how to show that $c_0$ is complete
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functional-analysis
metric-spaces
banach-spaces