This technical note presents a new proof of the circle criterion for multivariate, discrete-time systems with time-varying feedback nonlinearities. A new proof for the multivariate Tsypkin criterion for time-invariant monotonic feedback nonlinearities is derived as well. Both integrator- and non-integrator systems are considered. The proofs are direct in the sense that they do not resort to any existing result in systems theory, such as Lyapunov theory, passivity theory or the small-gain theorem. Instead, the proofs refer to the asymptotic properties of block-Toeplitz matrices. One major advantage of the new proof is that it elegantly handles integrator systems without resorting to loop transformation/pole shifting techniques. Additionally, less conservative stability bounds are derived by making stronger assumptions on the sector bound conditions on the feedback nonlinearities. In particular, it is exemplified how this technique relaxes stability conditions of (i) a model predictive control (MPC) rule and (ii) an integrator system.