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Math Help - Set, monotone sequence, inf

  1. #1
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    Set, monotone sequence, inf

    Hi, I have a question about the following proposition:


    Let S R be a nonempty bounded set. Then there exists a monotone sequence {xn} such that xn S for each n N and limxn = inf (S).


    How would one go about proving this? I know that a monotone sequence means that xn is greater or equal to xn+1 (or less than or equal to, but in this situation, because we want inf(S), we want monotone decreasing, right?).

    Help?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by seven.j View Post
    Hi, I have a question about the following proposition:


    Let S R be a nonempty bounded set. Then there exists a monotone sequence {xn} such that xn S for each n N and limxn = inf (S).


    How would one go about proving this? I know that a monotone sequence means that xn is greater or equal to xn+1 (or less than or equal to, but in this situation, because we want inf(S), we want monotone decreasing, right?).

    Help?
    If \inf S\in S choose the sequence to be the constant sequence x_n=\inf S, otherwise \inf S is a limit point of S. So, try choosing arbitrary neighborhoods of S and choosing arbitrary points in them.
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