Results 1 to 2 of 2

Math Help - Cauchy Sequence in Metric Space

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    63

    Cauchy Sequence in Metric Space

    Let \{x_n\} be a Cauchy sequence in X. Show that there exists a subsequence \{x_{n_{k}}\} of \{x_n\} such that \sum_{k=1}^{\infty} d(x_{n_{k}},x_{n_{k+1}}) < \infty.

    I can only say that since \{x_n\} is a Cauchy sequence, \{x_n\} is bounded, and since \{x_n\} is bounded, by Bolzano-Weierstrass Theorem, the sequence has a convergent subsequence. So how do I go about continuing from here?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: Cauchy Sequence in Metric Space

    You can construct by induction the sequence \{n_k\} such that d(x_{n_k},x_{n_{k+1}})\leq 2^{-k}.
    Note that we cannot necessary extract a converging subsequence of a bounded sequence, for example if X is an infinite dimensional vector space.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. All Cauchy Sequences are bounded in a metric space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 1st 2011, 03:18 PM
  2. Cauchy sequence convergence in a general metric space
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 29th 2010, 09:31 PM
  3. Cauchy Sequence in a Metric Space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 8th 2010, 03:05 AM
  4. metric space: convergent sequence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 11th 2009, 11:40 AM
  5. Cauchy Sequence in metric space problem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 17th 2007, 10:40 AM

Search Tags


/mathhelpforum @mathhelpforum