Preceding is Approximating Relation

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

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

Proof
Let $x \in S$.

Define $\RR := \mathord\preceq$.

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


 * $x^\preceq = x^\RR$

where:


 * $x^\preceq$ denotes the lower closure of $x$
 * $x^\RR$ denotes the $\RR$-segment of $x$

Thus by Supremum of Lower Closure of Element:
 * $x = \map \sup {x^\preceq} = \map \sup {x^\RR}$

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