Finite channel information maxima are attained

Regard the self-adjoint endomorphisms of $A$ as a real vector space of dimension $d^2$, where $d=\dim A\ge 1$, using an orthonormal basis of $A$ and the real diagonal and real/imaginary off-diagonal matrix coordinates. The state set is nonempty because $I_A\mathrm{/}d$ is a state. It is closed: trace one and the self-adjoint matrix-coordinate equalities are closed conditions, and for each fixed $x\in A$, the quadratic form condition $⟨\rho x,x⟩\ge 0$ is closed by finite-dimensional continuity; [F1]–[F2] show their intersection is exactly the state set. It is bounded in the coordinate norm on Hermitian matrices: each eigenvalue lies in $[0,1]$, so the operator norm is at most one; for standard basis vectors $e_i,e_j$, [F2] gives $\mathrm{\mid }⟨\rho e_j,e_i⟩\mathrm{\mid }\le 1$, hence all $d^2$ real matrix coordinates are bounded. The operator and coordinate norms induce the same topology by [F5], so [F6] makes the state set compact in operator norm.

The channels $\mathcal{N}$ and ${\mathcal{N}}^c$ are finite-dimensional linear maps and hence continuous in operator norm by [F5]. Their outputs on states are states. By [F3]–[F4], $\rho \mapsto S(\mathcal{N}(\rho ))-S({\mathcal{N}}^c(\rho ))$ is continuous; adding the continuous function $S(\rho )$ shows that the second displayed objective is continuous as well.

Apply [F7] to each of these two continuous real-valued functions on the nonempty compact state space from step 1.1. Each therefore attains its maximum, as asserted.