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Math Help - limits in a metric space question

  1. #1
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    limits in a metric space question

    The Question:

    Show that if x_n -> x in a metric space (M,d), then for any a in M, d(x_n,a) -> d(x,a).


    Where I am struggling:

    I understand that x_n->x => d(x_n,x) -> 0

    Then I tried relating this to the triangle inequality:

    d(x_n,a) <= d(x_n,x) + d(x,a)
    lim[ d(x_n,a)] <= 0 + d(x,a)
    d(x_n,a) -> c <= d(x,a)

    I don't know how to show that c is strictly equal to d(x,a) and not less than d(x,a).

    Thank you for any help that can be offered.
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  2. #2
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    Quote Originally Posted by rgriss1 View Post
    The Question: Show that if x_n -> x in a metric space (M,d), then for any a in M, d(x_n,a) -> d(x,a).
    There is a wonderfully useful theorem: If \left\{ {a,b,x} \right\} \subseteq M then \left| {d(a,x) - d(x,b)} \right| \leqslant d(a,b)
    Here we can say that \left| {d(x_n ,a) - d(a,x),} \right| \leqslant d(x_n ,x).
    We are almost done.
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  3. #3
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    Haha, yes. Thank you so much.

    I actually started messing around with the inequality after I posted this and remembered that theorem.

    But, again, thank you for confirming that I eventually found the right direction. :-)
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