We present a new approach to the stability analysis of finite receding horizon control applied to constrained linear systems. By relating the final predicted state to the current state through a bound on the terminal cost, it is shown that knowledge of upper and lower bounds for the finite horizon costs is sufficient to determine the stability of a receding horizon controller. This analysis is valid for receding horizon schemes with arbitrary positive-definite terminal weights and does not rely on the use of stabilizing constraints. The result is a computable test for stability, and two simple examples are used to illustrate its application.