Extensive approximate dynamic programming (ADP) algorithms have been developed based on policy iteration. For policy iteration based ADP of deterministic discrete-time nonlinear systems, existing literature has proved its convergence in the formulation of undiscounted value function under the assumption of exact approximation. Furthermore, the error bound of policy iteration based ADP has been analyzed in a discounted value function formulation with consideration of approximation errors. However, there has not been any error bound analysis of policy iteration based ADP in the undiscounted value function formulation with consideration of approximation errors. In this paper, we intend to fill this theoretical gap. We provide a sufficient condition on the approximation error, so that the iterative value function can be bounded in a neighbourhood of the optimal value function. To the best of the authors' knowledge, this is the first error bound result of the undiscounted policy iteration for deterministic discrete-time nonlinear systems considering approximation errors.