Every ray from a point in the interior of a compact convex set intersects with its boundary.

Question

Let $C$ be the compact convex set of $\mathbb{R}^n$ and let $R$ be the ray, and $O$ be the initial point of $R$. Then $R \cap C$ is a line segment(!). Let the other endpoint of $R \cap C$ $P$. if $P \in \operatorname{int}(C)$ then there is $B_\epsilon(P) \subset C$ so we can extend the line segment, leading to a contradiction. So $P \in \partial C$. How can we prove (!) ?

Answer 1

Since both $R$ and $C$ are convex, $R\cap C$ is convex too. Besides, $O\in R\cap C$. But the only convex closed subsets of $R$ are the closed intervals of $R$. If $R\cap C=R$, then $C$ would not be compact (since $R$ is unbounded). And the only intervals of $R$ to which $O$ belongs are those that go from $O$ to a point $P\in R$.