We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for input-constrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.