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


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    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).
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