Question about supremum and infimum

  • Thread starter hkcool
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  • #1
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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?
 
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Answers and Replies

  • #2
237
5
For any ε > 0 then yes, that's true.
 
  • #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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