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.