Suppose that there are infinitely many firms, all of which have the same cost function $C$. Assume that $C(0) = 0$, and for $q > 0$ the function $C(q)/q$ has a unique minimizer $\underline{q}$; denote the minimum of $C(q)/q$ by $\underline{p}$. Assume that the inverse demand function $P$ is decreasing. Show that in any Nash equilibrium the firms’ total output $Q^∗$ satisfies $$P(Q^∗ + \underline{q}) ≤ \underline{p} ≤ P(Q^∗)$$ that is, the price is at least the minimal value $\underline{p}$ of the average cost, but is close enough to this minimum that increasing the total output of the firms by $\underline{q}$ would reduce the price to at most $\underline{p}$. Hint: To establish these inequalities, show that if $P(Q^∗) < \underline{p}$ or $P(Q^∗ + \underline{q}) > \underline{p}$ then $Q^∗$ is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit.
So far I managed to proof the second part, $\underline{p} ≤ P(Q^∗)$, but I'm stuck at the first. All I managed to show is that if there is an equilibrium such that $P(Q^∗ + \underline{q}) > \underline{p}$, then every firm is making a profit (if they're not then they can always increase/decrease their output to $\underline{q}$), but other then that I'm stuck.