The notion of convergent systems is a powerful tool both in the analysis and synthesis of nonlinear systems. Sufficient conditions for convergence have been under investigation for smooth systems and for classes of non-smooth switching systems in the literature. In this paper, we look at a very particular class of non- smooth systems, namely complementarity systems. These systems have the capability of capturing the non-smooth dynamics of various interesting applications from different fields of engineering. The main contribution of this paper is to show that a linear complementarity system is convergent if the underlying linear dynamics possesses a certain positive realness property.