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Question about supremum and infimum

  1. Sep 30, 2012 #1
    Say we have [itex]a = \sup \{ a_{1}, a_{2}, a_{3}, ... \}[/itex]. Then does this mean we can find some [itex]a_{n} \in \{ a_{1}, a_{2}, ... \}[/itex] such that

    [tex]|a - a_{n}| < \varepsilon[/tex]

    ? My reasoning is that since a (the supremum) is the least upper bound of the set, we have to be able to find some member of the set that is arbitrarily close to a otherwise a wouldn't be a supremum anymore. Is this true?
    Last edited: Sep 30, 2012
  2. jcsd
  3. Sep 30, 2012 #2
    For any ε > 0 then yes, that's true.
  4. Oct 1, 2012 #3
    Well, if the the sup in the set iself then epsilon can indeed be zero.
    You may also find it interesting to think about finite sets.
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