Does every set with a supremum contain a monotone net converging to that supremum?

Question

It's well known that if $U \subset \mathbb{R}$ is bounded, then there exists a monotone increasing sequence $(x_{n})^{\infty}_{n=1}$ converging to $sup(U)$.

My question is: Let $X$ be a lattice, and let $U \subseteq X$ be a totally ordered subset (chain). Does there exist a monotone increasing net $(x_{\alpha})_{\alpha \in D}$ such that $sup_{\alpha}x_{\alpha} = sup(U)$?

Answer 1

Sure. Take $D:=U$ and define $x_\alpha:=\alpha$.