Upper Set is Convex

Theorem
Let $\struct {S, \preceq}$ 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:
 * $a \in T$
 * $a \preceq b$
 * $T$ is an upper set

it follows that:
 * $b \in T$

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

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

Also see

 * Lower Set is Convex