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This paper focuses on the stability of receding horizon control of general nonlinear systems. We employ a recent result on the convergence of the cost functional and we show that the cost induces a Lyapunov function along the receding horizon trajectories. The result requires a notion of detectability of nonlinear systems and a condition on the horizon length. Contrary to the usual approaches, we do not impose end-point constraints or special terminal weighting. In the particular linear setting, we show that the requirements imposed in the general case are reduced to the standard detectability and stabilizability conditions.