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Math Help - Prove a metric space

  1. #1
    Member thaopanda's Avatar
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    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).
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  2. #2
    Eater of Worlds
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    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}.
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  3. #3
    Member thaopanda's Avatar
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    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....
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  4. #4
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    Quote Originally Posted by thaopanda View Post
    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.
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