= Solution
For any region $B_n$ with $P^{\otimes n}(B_n)\geq1-\varepsilon$, let
$$
C_n=\left\{x_1^n:n^{-1}\log\frac{P^{\otimes n}(x_1^n)}{Q^{\otimes n}(x_1^n)}
\leq D(P\Vert Q)+\delta\right\}.
$$
The <weak law of large numbers> gives $P^{\otimes n}(C_n)\to1$, so $P^{\otimes n}(B_n\cap C_n)\geq1-\varepsilon-o(1)$. Therefore
$$
\beta_n\geq Q^{\otimes n}(B_n\cap C_n)
\geq e^{-n(D(P\Vert Q)+\delta)}\{1-\varepsilon-o(1)\}.
$$
Thus $\limsup-n^{-1}\log\beta_n\leq D(P\Vert Q)+\delta$; let $\delta\downarrow0$.
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