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Show seq. [itex] x_n [/itex] with [itex] |x_{n+1} - x_n| < \epsilon [/itex] is Cauchy

  1. Dec 5, 2011 #1
    1. The problem statement, all variables and given/known data

    The problem is longer but the part I'm stuck is to show that [itex] \{x_n\} [/itex] is convergent (I thought showing it is Cauchy) if I know that for all [itex] \epsilon > 0 [/itex] exists [itex] n_0 [/itex] such that for all [itex] n \geq n_0 [/itex] I have that
    [itex] |x_{n+1} - x_n| < \epsilon[/itex]

    2. Relevant equations

    A sequence is Cauchy if for all [itex] \epsilon > 0 [/itex] and for all [itex] n,m \geq n_0 [/itex] one has
    [itex] |x_m - x_n| < \epsilon [/itex]


    3. The attempt at a solution

    I called [itex] m = n+p [/itex] (for [itex] p [/itex] an arbitrary positive integer)
    Then
    [itex] |x_m - x_n| = |x_{n+p} - x_n|[/itex]
    But (and I think there is some mistake here):
    [itex] |x_{n+1} - x_n| < \epsilon/p [/itex]
    [itex] |x_{n+2} - x_{n+1}| < \epsilon/p [/itex]
    [itex] \vdots [/itex]
    [itex] |x_{n+p} - x_{n+p-1}| < \epsilon/p [/itex]

    So
    [itex] |x_{n+p} - x_n| < \underbrace{|x_{n+1} - x_n|}_{< \epsilon/p} + \underbrace{|x_{n+2} - x_{n+1}|}_{< \epsilon/p} + \ldots + \underbrace{|x_{n+p} - x_{n+p-1}|}_{< \epsilon/p} < \epsilon [/itex]

    Any help on why it's wrong (if it is) and how to solve it correctly?
    Thanks!
     
  2. jcsd
  3. Dec 5, 2011 #2

    Office_Shredder

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    Re: Show seq. [itex] x_n [/itex] with [itex] |x_{n+1} - x_n| < \epsilon [/itex] is Ca

    This isn't true. For example the sequence
    [tex] x_n = \sum_{i=1}^{n} 1/i[/tex]
     
  4. Dec 5, 2011 #3
    Re: Show seq. [itex] x_n [/itex] with [itex] |x_{n+1} - x_n| < \epsilon [/itex] is Ca

    You are right, thanks.

    I suppose I have to write the full problem: Given [itex] \{x_n\} [/itex] a sequence of real numbers, and [itex] S_n = \Sigma_{n=1}^n |x_{k+1} - x_k| [/itex], with [itex] S_n [/itex] bounded, prove that [itex] \{ x_n \} [/itex] converges.

    My attempt at a proof:
    Clearly [itex]\{ S_n \} [/itex] converges as it is a series of positive terms and it is bounded.
    So I define [itex] a_k = | x_{k+1} - x_k| [/itex], and now I know that [itex] \lim_{n \to \infty} a_n = 0 [/itex] (because the series [itex] S_n [/itex] converges).

    From there I really didn't know how to continue, I thoght proving [itex] x_n [/itex] was Cauchy, but didn't work. Any help?
     
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