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Thread: Sequence proof

  1. #1
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    Sequence proof

    I need help with the following proof:

    Let $\displaystyle (x_{n})_{n}$ be a sequence of real numbers such that $\displaystyle x_{n} + x_{n+1}$ converges and $\displaystyle x_{n}x_{n+1}$ converges. Show that $\displaystyle (x_{2n}) $converges
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well clearly then $\displaystyle a_n = x_{2n}+x_{2n+1}$ and $\displaystyle b_n = x_{2n}x_{2n+1}$ converge. Therefore $\displaystyle \frac{a_n\pm\sqrt{a_n^2-4b_n}}{2} = \{x_{2n},x_{2n+1}\}$ both converge.
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  3. #3
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    why is it that we can just pass it though the quadratic formula?
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