Solution
= Solution
Every word in $C_n(r)$ has probability at least $2^{-nr}$, so normalization gives $|C_n(r)|\leq2^{nr}$. Choose $H(U)<r<R$ and then $\varepsilon<r-H(U)$. Every $\varepsilon$-typical word satisfies $p(u^n)\geq2^{-n(H(U)+\varepsilon)}\geq2^{-nr}$, hence $T_\varepsilon^{(n)}\subseteq C_n(r)$ and $\Pr[C_n(r)]\to1$. Injectively encoding $C_n(r)$ and using a default codeword outside it gives rate at most $R$ and vanishing error.