A complete proof is given of global convergence to zero of the prediction error and asymptotic optimality for a direct adaptive controller is established based on multistep quadratic cost minimization with a receding horizon philosophy. The only substantial assumptions are that the controller designed when the parameters are known, stabilizes the plant and the exact knowledge of the firstNy(Ny-optimization horizon) impulse response coefficients of the plant. The technical hurdles for the development of a comprehensive convergence theory of this type of controller are also highlighted.