Open Ray is Dual to Open Ray

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $R$ be an open ray in $\left({S, \preceq}\right)$.

Then $R$ is an open ray in $\left({S, \succeq}\right)$, where $\succeq$ is the dual ordering of $\preceq$.

Proof
By the definition of open ray, there is some $p \in S$ such that:
 * $R = {\dot\uparrow}_\preceq p$ or $R = {\dot\downarrow}_\preceq p$

where ${\dot\uparrow}_\preceq p$ and ${\dot\downarrow}_\preceq p$ are the strict upper closure and strict lower closure of $p$ relative to $\preceq$, respectively.

By Strict Lower Closure is Dual to Strict Upper Closure, $R = {\dot\downarrow}_\succeq p$ or $R = {\dot\uparrow}_\succeq p$.

Thus $R$ is an open ray in $\left({S, \succeq}\right)$.