If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval?
This is from Exercise 7 in p. 92 in Munkres's Topology.
Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray.
But I cannot start my proof because I cannot use $\sup$ or $\max$ functions.
What should I do?
What about the set $\{p\in\Bbb Q:p^2<2\}$? It’s a non-trivial convex subset of $\Bbb Q$; does it meet Munkres’ definition of interval or ray?