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Sequences and existence of limit 2

  1. Nov 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Let an be a bounded sequence and bn such that

    the limit bn as n→∞ is b and

    0<bn ≤ 1/2 (bn-1)

    Prove that if:

    an+1 ≥ an - bn,

    then

    lim an
    n→∞

    exists.

    2. Relevant equations



    3. The attempt at a solution


    as 0<bn ≤ 1/2 (bn-1) the sequence bn is decreasing.
    thus its limit, b, is 0 (can i assume that?)

    Then, assuming an converges to a limit, say p, the equation is:

    p ≥ p - b where b=0

    p ≥ p is true for every p.

    Then, an is constant and therefore converges.

    Does is it work like that?
     
  2. jcsd
  3. Nov 12, 2012 #2

    SammyS

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    You can't assume that an converges. That's what you need to prove.
     
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