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In a Dense Order there is a Net Converging to Supremum

Last updated Nov 6, 2022

# 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. $\blacksquare$

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