This paper presents a relatively simple proof of the maximum principle. The main objective has been to obtain a proof, similar to that due to Halkin, but replacing the use of Brouwer's fixed point theorem by an easily proven contraction mapping theorem. The first use of Brouwer's fixed point theorem, in proving the convexity of the reachable set. of the linearized system, is eliminated by a simple, constructive proof of Lyapunov's theorem (requiring a slightly more general class of admissible controls than the piecewise continuous class employed by Halkin). The second use, in proving the fundamental lemma, is eliminated by a more powerful Balayage result and the use of an additional property of the linearized model of the non-linear system, permitting a contraction mapping theorem to be employed.