Solutions of continuous ODEs obtained as the limit of solutions of Lipschitz ODEs

One method of proving the existence of solutions for ODEs (x) over dot = f( x), where f is continuous, is to approximate f by a sequence of Lipschitz f unctions f, for which standard existence results can be applied. This short paper shows conversely that, in a phase space that is not two-dimensional, for each solution of (x) over dot = f(x) (such solutions may not be unique ) there is a sequence of Lipschitz functions f(n) which approximate f and w hich have solutions which converge to the chosen limit.