Lower Section is Convex

Theorem

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

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

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 section

it follows that:

$b \in T$

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

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

$\blacksquare$

Also see

• Upper Section is Convex