I have to show that the real numbers equipped with the metric

d(x,y)=|arctan(x)−arctan(y)| is an incomplete metric space.Certainly, I have to search for a Cauchy sequence of real numbers with respect to given metric that must not be convergent. But I am unable to figure out that.

Can anybody help me with this.Thanks for helping me.

**Answer**

Sorry for reviving such an old problem…

Anyways, what is important here is that arctan is a bijection from R to (−π/2,π/2), and it is an isometry if we give (−π/2,π/2) the metric it carries as a subspace of R with the usual metric. If f:X→Y is a surjective isometry then the Cauchy sequences in Y are the images of Cauchy sequences under f, and the convergent sequences in Y are the images of convergent sequences under f, so Y is complete if and only if X is. Thus (R,d) is complete if and only if (−π/2,π/2) with the metric dist(x,y)=|x−y| is complete. But (−π/2,π/2) is not complete since {π/2−1/n}∞n=1 is Cauchy and does not converge in (−π/2,π/2).

**Attribution***Source : Link , Question Author : Srijan , Answer Author : Adam*