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 sla | | and is symmetric with respect to the plane , then it intersects this plane in a countable union of strictly convex closed curves.