How to Prove Incompleteness and Completion in Metric Spaces?

[GreenBeret]

In the metric space (\mathbb R, d)

1) d(x,y) = |{tan}^{ - 1}(x) - {tan}^{ - 1}(y)| ,where x,y are real numbers .

2) d(x,y) = |{tan}^{ - 1}(x) - {tan}^{ - 1}(y)|, where x,y are real numbers .

Show that (\mathbb R, d) w.r.t (1) and (2) are incomplete metric space . Also, what is the completion space of both w.r.t. (1) and (2).

I appreciate any help.

 

[quasar987]

(1) and (2) are the same metric

 

[adriank]

Show that (1, 2, 3, ...) is a Cauchy sequence under the given metric that does not converge.