Let $(L,\prec)$ be dense linear ordering without end points. Show that $(L,\prec)$ is separable if and only if $(L,\prec)$ is order isomorphic to a sub-ordering of the real line $(\mathbb{R}, <).$
I know that if $(L,\prec)$ is separable then $(L,\prec)$ is a countable dense linear order without end points. Then by Baire Category Theorem, there exists a proper Dedekind cut $D \subseteq L$ such that $D$ avoids every $X \in \mathcal F$ where $\mathcal F$ is a countable family of closed nowhere dense subset in $(L,\prec)$. But I do not know how to proceed from here.