I would like to know if the following reasoning is correct:
If we have $a, b \in \mathbb{R}$ and $a < b$ then $\exists c \in \mathbb{R}$ such that $a < c < b$. From this, it follows that if $A$ is a set of real numbers then $\sup(A) = x$ for some $x \in A$. For if this were not the case, we would have $\sup(A) > x \quad \forall x \in A$. Hence we would have some $y$ such that $x \le y < \sup(A)$, which is clearly a contradiction.