 Solution

For fixed $\epsilon >0$, the asymptotic equipartition property gives $\mathrm{Pr}(X^n\notin A^{n,\epsilon })\to 0$. The strong law of large numbers applied to $X_i$, which are bounded because their numerical alphabet is finite, gives $\mathrm{Pr}(X^n\notin B^{n,\epsilon })\to 0$. The union bound therefore yieldsSince $\delta >0$, eventually $\mathrm{Pr}(X^n\in S_n)\ge 2^{-\delta }$. Each word in $S_n$ has probability mass at most $2^{-n(H(p)-\epsilon )}$, soRearranging provesfor sufficiently large $n$. The key is that the factor $2^{-\delta }$ is a fixed number below one, however small $\delta$ is. This is an instance of typical sets with an additional high-probability constraint.