Strictly feasible DMC input laws achieve every lower mutual-information rate

If $R=0$, the one-message code that repeats a minimum-cost symbol has zero error and cost $c_{\min }\le \mathrm{\Gamma }$. Suppose $R>0$. If $\mathrm{\Gamma }>c_{\min }$, [F1] gives an input law $P_X$ with $\bar{c}(P_X)<\mathrm{\Gamma }$ and $I(P_X;W)>R$. If $\mathrm{\Gamma }=c_{\min }$, choose an attaining law from [F1]. Feasibility gives ${\sum }_x{P}_X(x)(c(x)-c_{\min })\le 0$; every summand is nonnegative, so this law is supported on minimum-cost symbols and its mutual information exceeds $R$.

Choose $\delta >0$ with $R+2\delta <I(P_X;W)$ and $M_n=\lceil e^{nR}\rceil$. For the IID input and product DMC, [F2] makes the miss probability at threshold $\ln (M_n-1)+n\delta$ tend to zero, since the threshold divided by $n$ tends to $R+\delta <I(P_X;W)$.

By [F3], the IID random-code ensemble has expected average error $a_n\to 0$. If $\mathrm{\Gamma }>c_{\min }$, the letter costs are bounded on the finite alphabet, so the variance of their $n$-letter average is $\mathrm{Var}(c(X_1))\mathrm{/}n$; [F5] implies the expected fraction $q_n$ of codewords whose average cost exceeds $\mathrm{\Gamma }$ tends to zero because $\mathbb{E}c(X_1)<\mathrm{\Gamma }$. Apply [F4] to obtain deterministic subcodes with every retained word feasible, error tending to zero, and rate loss tending to zero. If $\mathrm{\Gamma }=c_{\min }$, every drawn input letter has minimum cost, so every random codeword has cost exactly $\mathrm{\Gamma }$ and no pruning is needed; [F3] supplies deterministic codes with vanishing error.