Lemma: Binomial coverage

If $K\sim \mathrm{Bin}(n,p)$, then

Proof. For fixed $p$, the counts satisfying $p<\ell (n,K;\zeta )$ form an upper-tail event whose probability is at most $\zeta$, by (eq:cpL) and monotonicity. The corresponding lower-tail argument proves the upper-endpoint statement. The endpoint definitions cover $k=0,n$.