Ray is Convex

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

Let $I$ be a open or closed ray.

Then $I$ is convex in $S$.

Proof
The cases for upward and downward-pointing rays are equivalent, so suppose WLOG that $U$ is an upward-pointing ray.

By the definition of a ray, there is an $a \in S$ such that:
 * $I = {\uparrow}a$ or $I = {\bar\uparrow}a$.

Suppose that $x, y, z \in S$, $x \prec y \prec z$, and $x, z \in I$.

Then $a \preceq x \prec y$, so $a \prec y$.

Therefore $y \in {\uparrow}a \subseteq I$.

Thus $I$ is convex.

Also see

 * Upper and Lower Closures are Convex