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A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform δ-persistency of excitation (Uδ-PE), is sufficient for uniform global asymptotic stability in certain situations. Uδ-PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is "definitely nonzero" where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.