Generalized limit ($\mathrm{liminf}{x}_n\le L(x)\le \mathrm{limsup}{x}_n$)

A generalized limit on real bounded sequences is a positive linear functional of operator norm one that extends the ordinary limit of a sequence. Such a functional can be chosen by the Hahn-Banach theorem: extend the limit on the convergent sequence space while dominating it by the sublinear functional $p(x)={\mathrm{limsup}}_n{x}_n$. Applying the same domination to $-x$ gives the displayed lower bound. A Banach limit additionally has invariance under the left shift on bounded sequences.