= Solution
An error implies that some $\theta\ne\theta^*$ has likelihood at least that of $\theta^*$. A finite <union bound> and part (b) therefore give
$$
\limsup_{n\to\infty}\frac1n\log
\mathbb P(\widehat\theta_n\ne\theta^*)
\leq-C^*(\Theta),
$$
where
$$
C^*(\Theta)=
\min_{\theta\ne\theta^*}
\inf_{R:\,D(R\Vert P_{\theta^*})\geq D(R\Vert P_\theta)}
D_e(R\Vert P_{\theta^*}).
$$
Every inner infimum is strictly positive by the argument in part (b), and the minimum of finitely many positive numbers is positive.
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