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The paper deals with the consensus convergence analysis of discrete-time single-integrator multi-agent systems (MASs) in switching networks. Here, we suppose that the switching network is composed of strongly connected digraphs. By adopting a time-varying quadratic Lyapunov function, we give the rigorous proof that the consensus of the single-integrator MASs can be achieved by either a constant-gain protocol or a decaying-gain protocol. This Lyapunov function based consensus analysis method is different from some existed ones for unbalanced switching networks (e.g., the property of products of indecomposable and aperiodic matrices) and provides a new perspective to study the consensus problem of MASs in unbalanced switching networks with uncertainties (e.g., additive noises).