Does $F$ being an extreme ray of a convex set $C$ imply that the direction of $F$ is in the recession cone?

Question

Let $C$ be a convex set, and let $F=x+\mathbb{R}^+_0s$, $s\in\mathbb{R}\setminus\{0\}$, be a facet of $C$, that is, an extreme ray. Does this imply $s$ being an element of the recession cone, i.e., is $$y+\mathbb{R}^+_0s\subseteq C,\quad y\in C?$$ If yes, how do I show it?

Answer 1

That's wrong. Take $C = \{(x,y) ~ | ~ y> 0 \} \cup \{(0,0)\}$