On the Existence of States Saturating the Bogomol'nyi Bound in N=4 Supersymmetry

We give an argument showing that in N=4 supersymmetric gauge theories there exists at least one bound state saturating the Bogomol'nyi bound with electric charge $p$ and magnetic charge $q$, for each $p$ and $q$ relatively prime, and we comment on the uniqueness of such state. This result is a necessary condition for the existence of an exact S-duality in N=4 supersymmetric theories.