# Prove a metric space

• Sep 25th 2009, 04:57 PM
thaopanda
Prove a metric space
need more help.... (Crying)

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

(a) Show that (R,d) is a metric space.
(b) Find $\displaystyle B_{\frac{\pi}{12}}$(1).
• Sep 25th 2009, 05:12 PM
galactus
Here is a start:

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

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

Think about $\displaystyle \frac{\pi}{2}$.
• Sep 27th 2009, 09:33 AM
thaopanda
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....
• Sep 27th 2009, 09:53 AM
Plato
Quote:

Originally Posted by thaopanda
Let d : R $\displaystyle \times$ R $\displaystyle \rightarrow$ 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 $\displaystyle d$ is a metric.
That is simple if you note that $\displaystyle \arctan(x)$ is an injection.