# Incomplete and completion

1. Nov 27, 2008

### 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.

2. Nov 27, 2008

### quasar987

(1) and (2) are the same metric

3. Nov 29, 2008

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