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Math Help - A Real Analysis Proof

  1. #1
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    A Real Analysis Proof

    I need some help with proving a theorem. I really appreciate any kind of help.

    Theorem: Suppose [an] is a convergent sequence and sigma un is a convergent positive series, then sigma ((an)^2)(un) is convergent.

    The n's in an and un are subscripts.
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  2. #2
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    Quote Originally Posted by straman View Post
    I need some help with proving a theorem. I really appreciate any kind of help.

    Theorem: Suppose [an] is a convergent sequence and sigma un is a convergent positive series, then sigma ((an)^2)(un) is convergent.

    The n's in an and un are subscripts.
    Maybe I understand it wrong but it seems true, and not really difficult... Since (a_n)_n is convergent, (a_n^2)_n is convergent as well, hence it is bounded: there is M such that |a_n^2|\leq M for all n. From there, we deduce 0\leq a_n^2 u_n\leq M u_n and you can conclude.
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