Cycle spinning is a widely used approach for improving the performance of wavelet-based methods that solve linear inverse problems. Extensive numerical experiments have shown that it significantly improves the quality of the recovered signal without increasing the computational cost. In this letter, we provide the first theoretical convergence result for cycle spinning for solving general linear inverse problems. We prove that the sequence of reconstructed signals is guaranteed to converge to the minimizer of some global cost function that incorporates all wavelet shifts.