A constructive and elementary proof of Reny's theorem

In a recent but well known paper, Reny proved the existence of Nash equilibria for better-reply-secure games, with possibly discontinuous payoff functions. Reny's proof is purely existential, and is similar to a contradiction proof : it gives non hint of a method to compute a Nash equilibrium in the class of games considered. In this paper, we adapt the arguments of Reny in order to obtain, for better-reply-secure games : an elementary proof of Nash equilibria existence, which is a consequence of Kakutani's theorem, and a " constructive " proof, in the sense that we obtain Nash equilibria as limits of fixed-point of well chosen correspondences.