In a Dense Order there is a Net Converging to Supremum
# Definition
Let $(X, \leq)$ be a Total Ordering with the Order Topology. Suppose $R \subset X$ is Nonempty and $\sup\limits R \in X$ exists. Then there exists a monotone Net $(x_{\alpha}){\alpha \in A} \subset R$ such that $x{\alpha} \to \sup\limits R$.
# Proof
Since $R$ is a Total Ordering, it is also a Directed Partial Ordering. Thus, we can view $R$ as the non-decreasing Net $(a_{a}){a \in R}$. Since $\sup\limits R$ exists, we know $a{a} \to \sup\limits R$ because Order-Preserving Nets Converge to their Supremum. $\blacksquare$