Finite Blaschke products, infinite Blaschke products, and their convex combinations are all in the closed unit ball of H∞. Therefore, one may wonder about the closure of these sets, either with respect to the usual norm or with respect to the topology of uniform convergence on compact sets. These investigations lead to several interesting results. In particular, we discuss the Caratheodory approximation theorem and the role it plays in the proof of Halmos conjecture and its generalization due to Berger and Stampfli.