In my textbook the following is presented as a theorem, which sometimes is also cited as definition of the supremum:
For S $\subseteq \mathbb{R}$, S non-empty, $\sigma$ = sup S iff
1. $\sigma$ is an upper bound for S
2. $\forall \epsilon >0\:\exists y \in S \; y>\sigma-\epsilon$
I think 2 is just stating that for any element less than $\sigma$, there is some element in S that is greater than that element. So can condition 2 be rewritten as:
$\forall x < \sigma \: \exists y \in S \; y>x $
Yes:
If $x<\sigma$, then $\epsilon=\sigma-x>0$, hence $\exists y\in S: y>\sigma-\epsilon=\sigma-\sigma+x=x$.
Conversely, let $\epsilon>0$. Then $x=\sigma-\epsilon<\sigma$, hence $\exists y\in S:y>x=\sigma-\epsilon$.