Upper Section is Convex

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

Let $T \subseteq S$ be an upper set.

Then $T$ is convex in $S$.

Proof
Let $a, c \in T$.

Let $b \in S$.

Let $a \preceq b \preceq c$.

Since $a \in T$, $a \preceq b$, and $T$ is an upper set, $b \in T$.

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

Also see

 * Lower Set is Convex