C($\mathbb{R}$) is complete under the discrete metric.

Question

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?

Answer 1

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.