1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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)
    [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]

    [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?
  2. jcsd
  3. Dec 5, 2011 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook