Consider a set $X$ such that $X$ does not have a lower bound, and it has a supremum $C = \sup X$. If I have some $a \in \mathbb{R}$, does $a < C$ mean that $a \in X$?
I understand that since $C$ is an upper bound, we have $x \leq C$ for all $x \in X$, but does the fact that $C$ is the 'smallest upper bound' imply that the above is true?
Edit: $X \subset \mathbb{R}$, my apologies for forgetting.
The answer is no (considering of course that $X\subset\mathbb{R}$). Here is a counter-example : let $X=\mathbb{Z}^-$, the set of all negative integers. Its supremum is $C=-1$, there is of course no lower bound. But you can see that $-1.5<C$ but obviously $-1.5\notin\mathbb{Z}^-$.