= Generalized limit
{title2=$\liminf x_n\leq L(x)\leq\limsup x_n$}
= Generalised limit
{synonym}
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> $p(x)=\limsup_n x_n$. Applying the same domination to $-x$ gives the displayed lower bound. A <Banach limit> additionally has invariance under the <left shift on bounded sequences>.
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