Lower Section is Convex
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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 $S$.
$\blacksquare$