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

    \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

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

    is convergent and evaluate its limit.

    \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

    x_{n+1}x_n=x_n+2 then by induction x_n>1
    Last edited by quepux; Jan 27th 2013 at 02:05 PM. Reason: SOLVED
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