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Math Help - Limit Proof

  1. #1
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    Limit Proof

    Could someone help me prove the following:

    If lim (an) = 0 and bn is a bounded sequence, prove lim (anbn) = 0.
    Both limits are to infinity.
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  2. #2
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    Quote Originally Posted by JimmyT View Post
    Could someone help me prove the following:

    If lim (an) = 0 and bn is a bounded sequence, prove lim (anbn) = 0.
    Both limits are to infinity.
    As b_n is bounded there exists a b>0, such that |b_n|<b.

    Also as lim_{n->infty} a_n=0, for any epsilon>0, there exits an N, such that
    |a_n|<epsilon for all n>N.

    Hence |a_n b_n|<epsilon b, for all n>N.

    So for any epsilon', set epsilon = epsilon' /b, then there exists an N
    such that |a_n| < epsilon, and also |a_n b_n|<epsilon' , hence:

    lim_{n -> infty} a_n b_n = 0.

    RonL
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  3. #3
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    Thanks a lot.
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