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Math Help - Convergent sequences

  1. #1
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    Convergent sequences

    let
    Last edited by Godisgood; November 1st 2010 at 06:07 AM.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Godisgood View Post
    Let X_n and y_n be two convergent sequence in the metric space (X,d) with X_n converging to x and y_n converging to y. show that d(x_n, y_n) converges to d(x,y)

    Thanks
    What about

    d(x_n,y_n) \le d(x_n,x)+d(x,y_n)\le d(x_n,x)+d(x,y)+d(y,y_n)

    This implies

    d(x_n,y_n)-d(x,y) \le d(x_n,x)+d(y,y_n)

    This should get you started. This is half of the inequality.
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  3. #3
    MHF Contributor chisigma's Avatar
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    That is true only if d(*,*) is continous in [x,y]...

    Kind regards

    \chi \sigma
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  4. #4
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    Do you know that \left| {d(x,y) - d(x_n ,y_n )} \right| \leqslant \left| {d(x,y) - d(x_n ,y)} \right| + \left| {d(x_n ,y) - d(x_n ,y_n )} \right|~~(\#1)

    There is a well-known theorem for metric spaces:
    for any points a,~b~\&, t \in S it is the case that \left| {d(a,t) - d(b,t)} \right| \leqslant d(a,b).

    Apply that to both terms on the right of ~(\#1).
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