Given an ordered set $X$, say a subset $Y$ of $X$ is convex in $X$ if for each pair of points $a<b$ of $Y$, the entire interval $(a,b)$ of points of $X$ lies in $Y$. An interval is of the form $(a,b),(a,b],[a,b),[a,b]$, and a ray is of the form $(a,+\infty),(-\infty,a),[a,+\infty),(-\infty,a]$.
Let $X$ be an ordered set. If $Y$ is a proper subset of $X$ that is convex in $X$, does it follow that $Y$ is an interval or a ray in $X$?
Well, suppose $Y$ has a smallest element $a$ and a largest element $b$. Then $[a,b]\in Y$, and so $Y=[a,b]$ and $Y$ is an interval.
What if $Y$ has no smallest element? So for each element $a\in Y$, there exists an element $b\in Y$ such that $b<a$. What then?
HINT: Let $X$ be the set of non-zero real numbers with the usual order, and let $Y$ be the set of positive real numbers.