  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.
