Preceding is Approximating Relation

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\preceq$ is an approximating relation on $S$.

Proof
Let $x \in S$.

Define $\mathcal R := \mathord\preceq$.

By definitions of lower closure of element and $\mathcal R$-segment:


 * $x^\preceq = x^{\mathcal R}$

where:


 * $x^\preceq$ denotes the lower closure of $x$
 * $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$

Thus by Supremum of Lower Closure of Element:
 * $x = \sup \left({x^\preceq}\right) = \sup \left({ x^{\mathcal R} }\right)$

Hence $\preceq$ is an approximating relation on $S$.