A Subgame Property of Stable Equilibria

Stable sets project onto stable sets of subgames in the following sense: stable sets consist of normal form perfect equilibria; for any given ϵ, we could therefore look at the set of strategy profiles in a subgame that are induced by the set of ϵ-perfect equilibria in a neighborhood of a stable set. Now take the limit, as ϵ goes to zero, of the closures of these sets. It is shown that the set so obtained contains a stable set of the subgame.