Lower Section is Convex

Theorem
Let $(S, \preceq)$ be an ordered set.

Let $T \subseteq S$ be a lower set.

Then $T$ is convex in $S$.

Proof
Let $a,c \in T$.

Let $b \in S$.

Let $a \preceq b \preceq c$.

Since $c \in T$, $b \preceq c$, and $T$ is a lower set, $b \in T$.

Since this holds for all such $a$, $b$, and $c$, $T$ is convex in $S$.

Also see

 * Upper Set is Convex