Convex Set of Ordered Set is not necessarily Interval

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

Let $C$ be a convex set of $S$.

Then it is not necessarily the case that $C$ is an interval of $S$.

Proof
Consider the open ray of $S$:
 * $R = \set {x \in S: a \prec x}$

for some $a \in S$.

From Ray is Convex, $R$ is a convex set of $S$.

But $R$ is not an interval of $S$.