Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Pair of convergent subsequences

  1. Mar 14, 2010 #1
    Let (xn) be a bounded sequence that diverges. Show that there is a pair of convergent subsequences (xnk) and (xmk), so that

    [tex]
    lim_{k\rightarrow\infty}
    [/tex] [tex]\left|x_{nk} - x_{mk}\right| > 0[/tex]​
     
  2. jcsd
  3. Mar 14, 2010 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: Convergence

    Any bounded sequence of real numbers has a convergent subsequence- let that be (xnk). Removing those from the sequence gives a new sequence that is still bounded. It must also have a convergent subsequence. Further, there must exist a subsequence that converges to something other than the limit of (xnk) (if all convergent subsequences converged to the same thing, the sequence itself would be convergent) . Let such a subsequence be (xmk).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Pair of convergent subsequences
  1. Idea of a subsequence (Replies: 26)

Loading...