This paper is concerned with convergence theorems and error bounds for approximate solutions of nonlinear problems, with particular applications to Urysohn integral equations. It is an abbreviated version of a more extensive projected sequel by P.M. Anselone, J. Davis and P.M. Prenter. The main goal is to clarify, simplify, and sharpen theorems and proofs on nonlinear operator approximation by Anselone [1], Atkinson [2], Moore [6,7] and Prenter [8]. More general related results have been obtained by Stummel [9, 10].