Operational capacity formula for a finite DMC

A maximal-error reliable sequence is also average-error reliable because ${\epsilon }_{\mathrm{a}\mathrm{v}}\le {\epsilon }_{\max }$. Thus [F3] gives $C_{\max }(W)\le C_{\mathrm{a}\mathrm{v}}(W)\le C(W)$.

By [F4], every $0\le R<C(W)$ is average-error achievable, so $C_{\mathrm{a}\mathrm{v}}(W)\ge C(W)$ after taking the supremum when $C(W)>0$. If $C(W)=0$, [F7] gives $C_{\mathrm{a}\mathrm{v}}(W)\ge 0$. The case split is exhaustive by [F6]; in either case $C_{\mathrm{a}\mathrm{v}}(W)=C(W)$.

If $C(W)>0$, fix any $0\le R<C(W)$. By [F4] there is an average-error reliable sequence at rate at least $R$. Apply [F5] to each code: the maximal errors tend to zero and the log message count loses at most $\ln 2$ at blocklength $n$, so the resulting rate is still at least $R$ in the limit. Thus $C_{\max }(W)\ge R$ for every such $R$, and hence $C_{\max }(W)\ge C(W)$. If $C(W)=0$, [F7] gives $C_{\max }(W)\ge 0=C(W)$. The cases are exhaustive by [F6], and step 1.1 gives equality.