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Math Help - Bounded and Convergence Proof Question

  1. #1
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    Bounded and Convergence Proof Question

    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If (a_{n}) is a bounded sequence with the property that every convergent subsequence of (a_{n}) converges to the same limit a \in \Re, then (a_{n}) must converge.

    Prove the Claim.
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    Quote Originally Posted by Janu42 View Post
    assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.
    Claim: If (a_{n}) is a bounded sequence with the property that every convergent subsequence of (a_{n}) converges to the same limit a \in \Re, then (a_{n}) must converge.
    Prove the lemma: that there cannot exist two disjoint closed intervals each containing infinitely many points of a_n. (Use the given and Bolzano-Weierstrauss)

    By Bolzano-Weierstrauss the set \left\{a_n:n\in \mathbb{N}\right\} has a limit point L.

    Use the lemma to prove: if  \varepsilon  > 0 then the interval \left[ {L - \varepsilon ,L + \varepsilon } \right] contains almost all the terms of a_n.
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    Quote Originally Posted by Janu42 View Post
    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If (a_{n}) is a bounded sequence with the property that every convergent subsequence of (a_{n}) converges to the same limit a \in \Re, then (a_{n}) must converge.

    Prove the Claim.
    I'm probably making this more complicated than it has to be

    Proof by contradiction

    Suppose (a_{n}) does not converge to a. Then, there exists an e>0 and a sequence of points (b_{n}) such that
    l b_{n} - al > e for all n=1,2,3....

    Look at the sequence (b_{n}). By Bolzano-Weierstrauss, it has a convergent subsequence (c_{n}), which is also a subsequence of (a_{n}). So (c_{n}) must converge to a. But l c_{n} - al > e for all n...contradiction.
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  4. #4
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Janu42 View Post
    I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

    Claim: If (a_{n}) is a bounded sequence with the property that every convergent subsequence of (a_{n}) converges to the same limit a \in \Re, then (a_{n}) must converge.

    Prove the Claim.
    I had posted a similar question, and got this response.

    check this thread out:

    http://www.mathhelpforum.com/math-he...bsequence.html
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