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Thread: Prove this sequence is contractive and evaluate its limit

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    Prove this sequence is contractive and evaluate its limit

    A sequence $\displaystyle \left \{x_n\right \}_{n\ge1}$ is called contractive iff there exists a $\displaystyle c\in \left.[0,1 \right )$ such that, for all $\displaystyle n\ge1$

    $\displaystyle \left |x_{n+2}-x_{n+1}\right| \ge c\left|x_{n+1}-x_{n}\right |$

    Any contractive sequence is Cauchy.

    Use the above definition to prove that the sequence defined by

    $\displaystyle x_1 = \alpha > 2, x_{n+1}=\displaystyle{x_n+2 \over x_n}$

    is convergent and evaluate its limit.

    $\displaystyle \left |x_{n+2}-x_{n+1}\right|=\displaystyle{2 \over x_{n+1}x_{n}}\left|x_{n+1}-x_{n}\right|$

    I couldn't go any further than this, any help will be appreciated.

    SOLVED

    $\displaystyle x_{n+1}x_n=x_n+2$ then by induction $\displaystyle x_n>1$
    Last edited by quepux; Jan 27th 2013 at 01:05 PM. Reason: SOLVED
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