Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 2 / c / Solution

An error implies that some $\theta \ne {\theta }^{\ast }$ has likelihood at least that of ${\theta }^{\ast }$. A finite union bound and part (b) therefore givewhereEvery inner infimum is strictly positive by the argument in part (b), and the minimum of finitely many positive numbers is positive.