# Math Help - Prove a metric space

1. ## Prove a metric space

need more help....

Let d : R $\times$ R $\rightarrow$ R be given by d(x,y) := |arctan x - arctan y|.

(a) Show that (R,d) is a metric space.
(b) Find $B_{\frac{\pi}{12}}$(1).

2. Here is a start:

$|tan^{-1}(x_{n})-tan^{-1}(m)|\leq |tan^{-1}(n)-\frac{\pi}{2}|+|tan^{-1}(m)-\frac{\pi}{2}|$

approaches 0 as $m,n\to \infty$

Think about $\frac{\pi}{2}$.

3. I don't really get where you got that equation. I figure as much as the triangle inequality that I need to prove, but I'm lost otherwise....

4. Originally Posted by thaopanda
Let d : R $\times$ R $\rightarrow$ R be given by d(x,y) := |arctan x - arctan y|.
(a) Show that (R,d) is a metric space.
To do that, you need to show that $d$ is a metric.
That is simple if you note that $\arctan(x)$ is an injection.