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Math Help - Proving a subsequence converges to S

  1. #1
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    Proving a subsequence converges to S

    Let (x_n) be a bounded sequence for each n \in N let s_n:=sup{ x_k:k> n} and S=inf{ s_n}. Show that there exists a sub sequence of (x_n) that converges to S.

    Ok so clearly s_1> s_2>....> s_n.

    Since it is decreasing and bounded then s_n converges to S. Futhermore there exists a subsequence of s_n that is convergent that much also converge to S. My only problem is I don't know how to relate this back to (x_n)
    Last edited by hockey777; September 24th 2008 at 01:02 PM.
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  2. #2
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    I take that back, there is still need for help.
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  3. #3
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    Quote Originally Posted by hockey777 View Post
    Ok so clearly s_1> s_2>....> s_n.
    This does not make sense, s_n are sets, how can they be "larger" or "smaller"?
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    This does not make sense, s_n are sets, how can they be "larger" or "smaller"?

    no s_1 is the first position in the sequence s_n, but I see where you think that as I messed up on my original post, I fixed it.
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