In this paper we develop a solution to the discrete-timeℋ∞output feedback control problem for linear time-varying (LTV) systems. The solution is developed along the strategy set up by Doyleet al. in 1989 and the main ingredient in its derivation is the extension of the well known bounded real lemma to a (discrete) time-varying context, developed by van der Veen and Verhaegen in 1995. This approach contributes to the conceptual simplicity, and hence to the accessibility, of the solution. Apart from that, we treat the infinite-horizon case for LTV system of non-uniform state dimension, and varying input and output dimension. Both situations can easily occur in practice, for example in multirate sampled data control systems. The algorithm that can be derived from the solution presented in then applied to theℋ∞output feedback of a dynamical system changing from one operation point in its operation envelope to another.