= Ultrafilter construction of a Banach limit
{title2=$L(x)=\lim_{n\to\mathcal U}\frac1n\sum_{j=1}^nx_j$}
A <nonprincipal ultrafilter> on the <natural numbers> gives a <Banach limit> by taking the <ultralimit> of the <Cesàro means> of each bounded sequence. Compactness of a closed bounded interval, or disk for complex sequences, supplies the limit. Finite intersections in the filter give linearity; nonnegative averages give positivity; constants give norm one. The <Cesàro theorem for convergent sequences> gives agreement with ordinary limits because the ultrafilter contains every cofinite set. For the <left shift on bounded sequences>, the two means differ by $(x_{n+1}-x_1)/n$, which tends to zero, proving shift invariance.
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