A simple proof of the nonrenormalization theorems in string theory is given. We prove (with the assumption that space-time supersymmetry is nonanomalous) that the space-time superpotential is not re-normalized by string loops. As corollaries of this result we argue that in string perturbation theory, Calabi-Yau vacua (or any other classical solution with space-time supersymmetry) are solutions of the quantum equations of motion with zero cosmological constant, massless particles remain massless, and Yukawa couplings are not renormalized, to all orders. PACS numbers: 11.17.+y, 11.30.Pb