# Every cofinal real closed subfield is dense?

Consider a real closed subfield $$F_0$$ of a real closed field $$F_1$$ that is cofinal. Here, "$$F_0$$ is cofinal in $$F_1$$" means "For each $$x \in F_1$$, there exists $$y \in F_1$$ such that $$x < y$$". Then, is $$F_0$$ order dense in $$F_1$$?

It need not be dense. For instance, consider the real-closed field $$F_0$$ of Puiseux series with real algebraic coefficients, and the real-closed field $$F_1$$ of Puiseux series with real coefficients. The natural inclusion $$F_0\subset F_1$$ is cofinal, but not dense since for instance we have $$(\pi - \varepsilon,\pi+\varepsilon) \cap F_0 = \varnothing$$ for every positive infinitesimal $$\varepsilon \in F_1$$.