Results 1 to 4 of 4

Math Help - Cauchy Sequences

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    39

    Cauchy Sequences

    Could anyone give me a specific example of Cauchy Sequence in Metric Space X that does not converge?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    537
    Thanks
    10
    Consider (0,\infty) with the hereditary topology from \mathbb{R} and check (1/n).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,363
    Thanks
    1311
    Or, let X be the set of rational numbers with the "usual metric", d(x,y)= |x-y|. Take the sequence 3, 3.1, 3.14, 3.1415, 3.14159, 3.141592, ... where the nth term is the decimal expansion of \pi to n-1 decimal places. It is easy to see that this sequence is a Cauchy sequence and that it does not converge in the rational numbers.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2009
    Posts
    1

    Thumbs up Cauchy Sequences

    Quote Originally Posted by HallsofIvy View Post
    Or, let X be the set of rational numbers with the "usual metric", d(x,y)= |x-y|. Take the sequence 3, 3.1, 3.14, 3.1415, 3.14159, 3.141592, ... where the nth term is the decimal expansion of \pi to n-1 decimal places. It is easy to see that this sequence is a Cauchy sequence and that it does not converge in the rational numbers.
    I'm new to the forum. I like this example, because it seems like something I may be able to use on a future test. I'm a grad student taking Real Analysis I and need a few tricks in my back pockets. Thanks!
    Terry
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cauchy Sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 10th 2010, 09:29 PM
  2. Cauchy Sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2009, 05:35 PM
  3. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 21st 2009, 08:59 PM
  4. More Cauchy Sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 26th 2008, 01:16 PM
  5. Cauchy sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 26th 2008, 05:56 AM

Search Tags


/mathhelpforum @mathhelpforum