Verification - does $x < \sup A$ necessarily mean $x \in A$? Suppose we have a non-empty and bounded above set $A$, and some $x \in \mathbb{R}$ such that $x < \sup A$.
Do we then have that $x \in A$?
Since we assume $\sup A$ exists, it follows that $A \subset \mathbb{R}$.
If we take $x = \sqrt{2}$ and define $A = \{...,-3,-2,-1,0,1,2\}$
Then $A$ is non-empty, bounded above and has $\sup A = 2$. Furthermore $x < \sup A$, but $x \notin A$.
I know this seems rather elementary, but I just wanted to make sure there was absolutely no flaw in my example.
Is there an even simpler case for which you could prove this?
thanks.

Just take $x<\inf A$. Then $x$ can't belong to A
