Every cofinal real closed subfield is dense?

Question

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 $F_1$?

Answer 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 $\varepsilon \in F_1$.