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Cauchy sequence

  1. Jan 27, 2014 #1
    1. The problem statement, all variables and given/known data

    Given:

    [tex]x_{n+1}=\frac{1}{3+x_n}[/tex]

    with
    [tex] x_1=1 [/tex]

    Show that:

    (1)

    [tex]|x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}|[/tex]

    and (2) x_n is Cauchy.

    2. Relevant equations



    3. The attempt at a solution
    I've tried different approaches (including induction) but the sequence isn't monotonically decreasing.
     
  2. jcsd
  3. Jan 27, 2014 #2

    Office_Shredder

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    Have you tried writing out [itex] |x_{n+1} - x_{n}| [/itex] using the definition [itex] x_j = 1/(3+x_{j-1}[/itex]?
     
  4. Jan 27, 2014 #3
    Got it ty.
     
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