The proof of this Lemma is more involved. See Bernheim (1984: Appendix A) for a proof. In order for a strategy \(s_i\) to not be strictly dominated, it is sufficient for \(s_i\) to be a best response to a belief, whatever that belief is, about the opponents’ choices. Admissibility requires something more: the strategy must be a best response to a belief that does not explicitly rule-out any of the opponents’ choices. Comparing these two Lemmas, we see that strict dominance implies weak dominance, but not necessarily vice versa. A strategy might not be a best response to any full-support probability measure while being a best response to some particular beliefs, those assigning probability one to a state where the player is indifferent between the outcome of its present action and the potentially inadmissible one.