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Math Help - subsequence converging to p...

  1. #1
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    subsequence converging to p...

    Let (an) be a sequence of real numbers, and let E = {an : n ∈ N} be the range of (an). Prove that (an) has a subsequence converging to p iff either p appears infinitely many times in (an) or p is an accumulation point of E.

    For this if and only if is it showing that either it appears infinitely many times or p is an accumulation point of E?

    To be perfectly honest I'm not sure how to even think about this one.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dfanforlife View Post
    Let (an) be a sequence of real numbers, and let E = {an : n ∈ N} be the range of (an). Prove that (an) has a subsequence converging to p iff either p appears infinitely many times in (an) or p is an accumulation point of E.

    For this if and only if is it showing that either it appears infinitely many times or p is an accumulation point of E?

    To be perfectly honest I'm not sure how to even think about this one.
    If p appears infinitely often you can just take the subsequence to be that point repeated. If not, then for every \varepsilon>0 there exists some point of the sequence different from p in B_\varepsilon(p) and so in particular B_{\varepsilon}\cap P\supsetneq\{p\}. Since \varepsilon was arbitrary the conclusion follows.
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