Find the supremum of $S:=\{x \in \Bbb{R} : x^2 < 4\}$
The set $S$ is equivalent to the set $(-\sqrt {4},\sqrt{4})$. Therefore $\sqrt{4}$ is an upper bound. We have, for sufficiently small $\epsilon >0$, that $\sqrt{4} - \epsilon /2>\sqrt{4}-\epsilon$ ,and hence $\sqrt{4}-\epsilon$ cannot be an upper bound. Therefore $Sup(S)=\sqrt{4}.$ Is this correct? I think implicitly we may assume that $\epsilon$ is always small enough in such cases.