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Limit superior question

  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data

    [itex]\{x_{n}\}\in\mathbb{R^{+}}[/itex] is a bounded sequence and [itex]r=\lim\sup_{n\rightarrow\infty}x_{n}[/itex]. Show that [itex]\forall\epsilon>0,\exists[/itex] finitely many x_{n}>r+\epsilon and infinitely many [itex]x_{n}<r+\epsilon[/itex].

    3. The attempt at a solution

    By definition of limit superior, [itex]r\in\mathbb{R}[/itex] is such that [itex]\forall\epsilon>0[/itex], [itex]\exists N_{\epsilon}[/itex] s.t. [itex]x_{n}<r+\epsilon, \forall n>N_{\epsilon}[/itex]. This would imply that any [itex]x>r+\epsilon/[itex] is an upper bound on [itex]\{x_{n}\}[/itex]. How do I show that there are finitely many such upper bounds? Is it because [itex]\{x_{n}\}[/itex] is a bounded sequence that there must only be finite [itex]x_{n}>r+\epsilon[/itex] ?
     
  2. jcsd
  3. Jan 21, 2012 #2

    Office_Shredder

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    I'm not sure what you mean here, but what you've written basically answers the first part of the question. If for [itex]n>N_{\epsilon}[/itex], [itex] x_n<r+\epsilon[/itex], then there are at most [itex]N_{\epsilon}[/itex] values of xn which are larger than [itex]r+\epsilon[/itex]
     
  4. Jan 22, 2012 #3
    I meant that "there are finitely many such [itex]x\in(x_{n})[/itex]". How do I get started proving that there are infinitely many [itex]x\in(x_{n})[/itex] s.t. [itex]x<r+\epsilon[/itex]?
     
  5. Jan 23, 2012 #4
    Suppose there were only finitely many [itex]x_n[/itex] such that [itex]x_n < r + \epsilon[/itex]. Would this in any way contradict the given facts? (Think about the definition of lim sup)
     
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