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Examine if the space $X=C(\mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not.

We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct a function $f(x_n),Cauchy$ such that $f(x_n)\rightarrow f(x_0)$ and also $f$ is continuous.

Is this gonna work?

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If you endow any set with that discrete metric, it becomes a complete metric space, because any Cauchy sequence is constant then, if $n$ is large enough.

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  • $\begingroup$ I don't know if i used the right word because i tranlated the problem from Greek. But it basically asks you to prove that C($\mathbb{R}$),$d_0$) is complete or not. $\endgroup$ – argiriskar Oct 11 '18 at 11:36
  • $\begingroup$ This post answers your question. In order for a metric space to be complete, every Cauchy sequence must converges (with respect to the metric defined on the space). Every Cauchy sequence in your space must be constant for $n$ large enough, so it converges. $\endgroup$ – nicomezi Oct 11 '18 at 11:42

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