Results 1 to 2 of 2

Math Help - Convergence

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    10

    Convergence

    Let (x_n) be a bounded sequence that diverges. Show that there is a pair of convergent subsequences (x_{nk}) and (x_{mk}) so that

    lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid > 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by yusukered07 View Post
    Let (x_n) be a bounded sequence that diverges. Show that there is a pair of convergent subsequences (x_{nk}) and (x_{mk}) so that

    lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid > 0
    The equation can equivalently be written \lim_k x_{n_k}\neq \lim_k x_{m_k} since these limits are assumed to exist.

    I guess you know that a bounded sequence has a convergent subsequence.

    Then choose a convergent subsequence (x_{n_k})_k, with some limit \ell. Using the fact that initial sequence does not converge, you can find \epsilon_0 and a subsequence (x_{m_k})_k such that |x_{m_k}-\ell|>\epsilon_0 for all k (I let you justify that); then extract a convergent subsequence from (x_{m_k})_k and conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: March 27th 2011, 07:42 PM
  2. Replies: 1
    Last Post: May 13th 2010, 01:20 PM
  3. Replies: 2
    Last Post: May 1st 2010, 09:22 PM
  4. dominated convergence theorem for convergence in measure
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 5th 2009, 04:06 AM
  5. Replies: 6
    Last Post: October 1st 2009, 09:10 AM

Search Tags


/mathhelpforum @mathhelpforum