Here $e_2^{(n)}=P^{\otimes n}(B_n^c)$. The <method of types> gives at most $(n+1)^{|A|}$ possible values of <type (information theory)>, and a type class $T_R$ has
$$
P^{\otimes n}(T_R)\leq2^{-nD(R\Vert P)}.
$$
Every <type (information theory)> in $B_n^c$ has $D(R\Vert P)>\delta$, so
$$
e_2^{(n)}
\leq(n+1)^{|A|}2^{-n\delta}.
$$
The polynomial prefactor has zero exponential rate. Therefore
$$
\limsup_{n\to\infty}\frac1n\log e_2^{(n)}\leq-\delta.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-224/2/d/solution