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Homework Help: Real analysis monotone subsequence

  1. Sep 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Let (Xn) be a sequence in R (reals). Then (Xn) has a monotone subsequence.

    2. Relevant equations

    Def: Monotone: A sequence is monotone if it increases or decreases.

    3. The attempt at a solution

    I know it has something to do with peak points...that is there are elements in (Xn) which are peak points (every element afterwards is smaller). There are either an infinite number of peak points (in which case the subsequence consists of the peak points) of finite. I am having a hard time grasping what the subsequence consists of if there are a finite number of peak points...
  2. jcsd
  3. Sep 18, 2007 #2


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    Homework Helper

    If the sequence is unbounded, the result is easy. If it is bounded, it has a convergent subsequence. See if you can make this into a monotone subsequence.
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