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Math Help - Prove that the sequence is increasing and bounded above

  1. #1
    MHF Contributor alexmahone's Avatar
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    Prove that the sequence is increasing and bounded above

    Define a sequence by

    a_{n+1}=\frac{a_n+1}{2}, n\ge 0

    Prove that if a_0\le 1, the sequence is increasing and bounded above.

    My solution:

    a_{n+1}-1=\frac{a_n+1}{2}-1=\frac{a_n-1}{2}

    So, a_0\le 1\implies a_n\le 1 for all n, ie the sequence is bounded above by 1.

    a_{n+1}-a_n=\frac{a_n+1}{2}-a_n

    =\frac{1-a_n}{2}

    We know that a_n\le 1 for all n. So, a_{n+1}\ge a_n and the sequence is increasing.
    Last edited by alexmahone; January 15th 2012 at 01:02 AM. Reason: Solved
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  2. #2
    Super Member girdav's Avatar
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    Re: Prove that the sequence is increasing and bounded above

    Show by induction that a_n\leq 1 for all n.
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