Properly embedded surfaces with constant mean curvature

In this paper we prove a maximum principle at infinity for properly embedded surfaces with constant mean curvature $H>0$ in the $3$-dimensional Euclidean space. We show that no one of these surfaces can lie in the mean convex side of another properly embedded $H$ surface. We also prove that, under natural assumptions, if the surface lies in the slab $|x_3|<1/2H$ and is symmetric with respect to the plane $x_3 = 0$, then it intersects this plane in a countable union of strictly convex closed curves.