Lemma: Coverage of the one-sided binomial endpoint

The endpoint above satisfies ${\mathrm{Pr}}_q\{q>q_U(X)\}\le \delta$. For zero observed errors,

Proof. For fixed $q$, the endpoint is monotone in $x$. The event $q>q_U(X)$ is a lower-tail event whose probability is at most $\delta$ by the defining tail equation and discreteness. At $x=0$, the equation is $(1-q_U{)}^n=\delta$.