As in the finite-dimensional case, the appropriate state feedback for the infinite-dimensional H-infinity disturbance-attenuation problem may be calculated by solving a Riccati equation. This operator Riccati equation can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems and consider the corresponding finite-dimensional disturbance-attenuation problems. We make the same assumptions required in approximations for the classical linear quadratic regulator problem and show that the sequence of solutions to the corresponding finite-dimensional Riccati equations converge strongly to the solution to the infinite-dimensional Riccati equation. Furthermore, the corresponding finite-dimensional feedback operators yield performance arbitrarily close to that obtained with the infinite-dimensional solution.