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Math Help - Using the Sandwich Theorem

  1. #1
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    Using the Sandwich Theorem

    Suppose that the sequence (a_n) converges to 0 and (b_n) is bounded. Use the Sandwich Theorem to prove that (a_n)(b_n) converges to 0.

    I know that, since (b_n) is bounded, it is convergent to a limit L. But I'm not sure how to use the Sandwich Theorem to prove the product converges to 0 - any help would be greatly appreciated!
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  2. #2
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    Re: Using the Sandwich Theorem

    Quote Originally Posted by sakuraxkisu View Post
    Suppose that the sequence (a_n) converges to 0 and (b_n) is bounded. Use the Sandwich Theorem to prove that (a_n)(b_n) converges to 0.
    I know that, since (b_n) is bounded, it is convergent to a limit L. But I'm not sure how to use the Sandwich Theorem to prove the product converges to 0 - any help would be greatly appreciated!
    That is a false claim.

    It is true that \exists B>0 such that (\forall n)[|b_n|\le B]~.

    If \epsilon>0 then (\exists N>0)(\forall n\ge N)\left[|a_n|<\frac{\epsilon}{B}\right].

    Now you finish.
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    Re: Using the Sandwich Theorem

    Hi, thank you for your reply. Sorry about the mistake! I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this? Thanks
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    Re: Using the Sandwich Theorem

    Quote Originally Posted by sakuraxkisu View Post
    Hi, thank you for your reply. Sorry about the mistake! I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this?
    There is no reason to use the Sandwich Theorem.
    If you must, I don't see right now.
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    Re: Using the Sandwich Theorem

    Quote Originally Posted by sakuraxkisu View Post
    I was wondering, would you perhaps be able to explain how the Sandwich Theorem can be used to prove this?
    Consider c_n=0 and d_n=B|a_n| where |b_n|\le B for all n. Show that c_n\le |a_nb_n|\le d_n.
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