Let $X$ be an ordered set in the order topology. Show that $\overline{(a,b)}\subseteq[a,b]$.
So this amounts to showing that anything outside $[a,b]$ is not a limit point of $(a,b)$. So suppose $x\not\in[a,b]$, and assume $x<a$. We have to show that there exists an open set $G$ containing $x$ that doesn't contain anything from $(a,b)$. How to do so?
HINT: Every open ray is an open set, where an open ray is a set of the form $(x,\to)$ or $(\leftarrow,x)$.