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Math Help - Cauchy Sequence Question

  1. #1
    Super Member Aryth's Avatar
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    Cauchy Sequence Question

    Prove Directly that, if \{a_n\}_{n=1}^{\infty} and \{b_n\}_{n=1}^{\infty} are Cauchy, so is \{a_nb_n\}_{n=1}^{\infty}.

    I cannot use the same proof that shows that if a converges to A and b to B, then ab converges to AB.
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  2. #2
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    Re: Cauchy Sequence Question

    Quote Originally Posted by Aryth View Post
    Prove Directly that, if \{a_n\}_{n=1}^{\infty} and \{b_n\}_{n=1}^{\infty} are Cauchy, so is \{a_nb_n\}_{n=1}^{\infty}.
    Here is the basic trick
    \begin{align*}|a_mb_m-a_nb_n| &\le |a_mb_m-a_nb_m|+|a_nb_m-a_nb_m|\\ &\le |a_m-a_n||b_m|+|a_n||b_m-b_n|   \end{align*}
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