For analyzing the stability of discrete-time systems containing a feedback nonlinearity, the Tsypkin criterion is the closest analog to the Popov criterion which is used for analyzing such systems in continuous time, Traditionally, the proof of this criterion is based upon input-output properties and function analytic methods. In this paper we extend the Tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof for this classical result.