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Math Help - converges to L

  1. #1
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    converges to L

    Since I am not that great with finding subsequences I need help with writing this proof:

    Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

    I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by redsoxnation View Post
    Since I am not that great with finding subsequences I need help with writing this proof:

    Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

    I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?
    Give us more than that.
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Quote Originally Posted by redsoxnation View Post
    Since I am not that great with finding subsequences I need help with writing this proof:

    Let {x_n be a bounded sequence and suppose that every convergent subsequence of {x_n} converges to L. Prove that {x_n} converges to L.

    I know that this statement does not guarantee that every subsequence converges. How would I go about the rest?


    The Bolzano-Weierstrass theorem states that every bounded sequence will have a convergent subsequence. So, let \epsilon>0 and let there exist an N such that |\{x_{n_k}\}-L|<\epsilon for n_k>N and \{x_{n_k}\} is a subsequence of \{x_n\}.....

    no idea where to go from there...any input?
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