Past exam of the functional Analysis and Banach and C*-Algebras course of the University of Oxford / 2008 / A / 2 / ii

The extension already agrees with the usual limit on $c$. Alternatively, applying the domination inequality to both $x$ and $-x$ givesWhen the sequence converges, both endpoints are its limit, proving $\phi (x)=\lim {\alpha }_n$. These bounds also give $\Vert \phi \Vert =1$ and positivity on nonnegative sequences, but do not by themselves assert shift invariance.