This paper presents a detailed discussion about the convergence properties of a variable structure controller for uncertain single-input-single-output nonlinear systems (SISO). The adopted approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies and external disturbances that can arise. The boundedness of all closed-loop signals and the convergence properties of the tracking error are analytically proven using Lyapunov's direct method and Barbalat's lemma. This result corrects flawed conclusions previously reached in the literature. An application of this adaptive fuzzy sliding mode controller to a second-order nonlinear system is also presented. The obtained numerical results demonstrate the improved control system performance.