Question about supremum and infimum

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The discussion centers on the concept of supremum (sup) in mathematical analysis, specifically addressing whether a supremum, denoted as a = \sup \{ a_{1}, a_{2}, a_{3}, ... \}, guarantees the existence of an element a_{n} in the set such that |a - a_{n}| < ε for any ε > 0. It is established that this is indeed true, as the supremum is defined as the least upper bound of the set, implying that elements of the set can be found arbitrarily close to the supremum. The conversation also touches on the implications for finite sets.

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hkcool
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Say we have a = \sup \{ a_{1}, a_{2}, a_{3}, ... \}. Then does this mean we can find some a_{n} \in \{ a_{1}, a_{2}, ... \} such that

|a - a_{n}| &lt; \varepsilon

? 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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For any ε > 0 then yes, that's true.
 
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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