Question about supremum and infimum

[hkcool]

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

$$|a - a_{n}| < \varepsilon$$

? 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?

 

[skiller]

For any ε > 0 then yes, that's true.

 

[Useful nucleus]

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.