Let M_0\subseteq M_1 be two real ordered fields where M_0 is dense in M_1. Then is the real closure of M_0 dense in the real closure of M_1?
Suppose there were a nonempty open set that intersected the closure of $M_1$ but not the closure of $M_0$. Take a point in that set and look at its minimal polynomial over $M_1$. A sufficiently small neighborhood of the coefficients of that polynomial should produce only polynomials that have a root in that set, since roots of polynomials are continuous. Choose a point in $M_0$ in each of those neighborhoods, look at the polynomial with those coefficients, and find a root in that open set.