Interval of Ordered Set is Convex

Theorem
Let $(S, \preceq)$ be a totally ordered set.

Let $I$ be an open, closed, or half-open interval in $S$.

Then $I$ is convex in $S$.

Proof
Any interval can be represented as the intersection of two rays.

Thus by Ray is Convex and Intersection of Convex Sets is Convex Set (Order Theory), $I$ is convex.