An application of Kleene's fixed point theorem to dynamic programming

In this note, we show that the least fixed point of the Bellman operator in a certain set can be computed by value iteration whether or not the fixed point is the value function. As an application, we show one of the main results of Kamihigashi (2014, "Elementary results on solutions to the Bellman equation of dynamic programming:existence, uniqueness, and convergence," Economic Theory 56, 251-273) with a simpler proof.