The full paper can be found here: http://www.dklevine.com/archive/refs4397.pdf. I refer here to the Lemma 2 in page 557.
My doubt is regarding the first statement of the proof: "Because reaction functions are nonincreasing and $0$ is a realization of $R^1(q)$, firm 1 must react to any quantity above $q$ by setting $0$ with probability $1$". Why is that so? If $0$ is a possible realization of $R^1(q)$ (if I understood correctly what the authors mean by realization -- since the reaction functions can be understood as random variables --, and maybe that is where I am troubled), what guarantees the fact that any small quantity above $q$ will make firm 1 set its reaction function equal $0$?