We present a new formulation of a convergence result for Lyapunov function candidates satisfying a differential inequality with integrable coefficients that often appears in adaptive control problems. Usually, Barbalat's Lemma is invoked, requiring boundedness of the time derivative of the Lyapunov function candidate which can sometimes be hard to establish. By connecting results from the literature, an alternative route avoiding Barbalat's Lemma is suggested. (C) 2018 Elsevier Ltd. All rights reserved.