Lower Set is Convex

Theorem
Let $\left({S, \preceq}\right)$ 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:
 * $c \in T$
 * $b \preceq c$
 * $T$ is a lower set

it follows that:
 * $b \in T$

This holds for all such $a$, $b$, and $c$.

Hence, by definition, $T$ is convex in $S$.

Also see

 * Upper Set is Convex