  We study convergence of nonlinear systems in the presence of an `almost
Lyapunov' function which, unlike the classical Lyapunov function, is allowed to
be nondecreasing---and even increasing---on a nontrivial subset of the phase
space. Under the assumption that the vector field is free of singular points
(away from the origin) and that the subset where the Lyapunov function does not
decrease is sufficiently small, we prove that solutions approach a small
neighborhood of the origin. A nontrivial example where this theorem applies is
constructed.
