Lemma 2 in Tirole and Maskin's Dynamic Oligpoly I (1988) 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 $0$?
The monotonicity result that they prove in their lemma 1 (p. 556) is stronger than you think: it states that whenever $q&gt;q'$, the support of $R_1(q)$ is "uniformly below" the support of $R_1(q')$. In other words, it is impossible to find any realizations $(r,r')$ of $R_1(q)$ and $R_1(q')$, respectively such that $r&gt;r'$.

The proof of this result relies on the idea that firms decisions are strategic substitutes in the Cournot duopoly model: the more firm 2 produces, the less firm 1 wants to produce. Thus, if firm 1 finds it optimal to produce an amount $r'$ with positive probability when firm 2 produces $q'$, it cannot find it optimal to produce a larger amount ($r&gt;r'$) when firm 2 itself produces more ($q&gt;q')$.

Consider therefore a production level $q$ by firm 2 such that $0$ is a realization of $R_1(q)$. Take $\epsilon&gt;0$ and suppose that there exists $r&gt;0$ that is a realization of $R_1(q+\epsilon)$. The condition
\begin{equation*}
r &gt; 0 \text{ and } q&lt;q+\epsilon
\end{equation*}
violates their lemma 1, since $r$ is a realization of $R_1(q+\epsilon)$ and $q$ is a realization of $R_1(q)$. This proves that $R_1(q+\epsilon)$ has all its weight on 0.