This paper studies relations between nonlinear H-infinity, control of discrete-time nonlinear system and H-infinity control of the linearized system. It is shown that an input-output discrete-time nonlinear system has a local L-2 gain gamma > 0, with internal stability, if its linearized system is asymptotically stable and has an L-2 gain strictly less than gamma. Using this fundamental result, it is further proved that the existence of a local solution to the problem of disturbance attenuation with internal stability for nonlinear systems is guaranteed by the solution of the corresponding H-infinity control problem for the linearized systems. All the results are established directly via the simple and intuitive linearization technique which provides, in contrast with the existing approach presented by Lin in 1993 and by Lin and Byrnes in 1996, a new insight and an appealing alternative to the discrete-time nonlinear H-infinity control problem.