Shift domination criterion for a Banach limit ($q\le \mathrm{limsup},q(x-Sx)\le 0$)

Suppose $q$ is a sublinear functional on the bounded real sequences, dominated by the limit superior and nonpositive on every shift difference $x-Sx$. The Hahn-Banach theorem then yields a Banach limit. Indeed $0\le q(x)+q(-x)$ implies $q(x)\ge -q(-x)\ge \mathrm{liminf}{x}_n$, so $q$ agrees with the ordinary limit on convergent sequences. Extend this linear limit functional by Hahn-Banach to a linear $\psi \le q$. Domination on $x-Sx$ and on its negative forces $\psi (x)=\psi (Sx)$. The two-sided limit bounds give positivity, norm one and agreement with ordinary limits. In finite-shift averaging constructions, the needed bound on shift differences follows by telescoping consecutive averages to $(x_{n+1}-x_{n+k+1})/k$.