need more help.... 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).
Follow Math Help Forum on Facebook and Google+
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 $\displaystyle \frac{\pi}{2}$.
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....
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.
View Tag Cloud