On completness of $\mathbb{R}$

Question

Why $\mathbb{R}$ with $d(x,y)=|\arctan{x}-\arctan{y}|$ is an incomplete metric space?

Answer 1

Remind the definition of completeness of a metric space: a metric space $(M,d)$ is complete if every Cauchy sequence over $(M,d)$ converges. However the sequence $\langle n \mid n=1,2,3,\cdots\rangle$ does not converge although it is Cauchy.

Some interesting fact is that your metric gives the same topology of that given by the standard topology. Especially, every convergent sequence under the standard metric also converges under your metric and vice versa. Hence your example shows completeness of a metric space is a property of a metric, not a property of a topology.