Solution
= Solution
The extension already agrees with the usual limit on $c$. Alternatively, applying the domination inequality to both $x$ and $-x$ gives
$$
\liminf_n\alpha_n\leq\varphi(x)\leq\limsup_n\alpha_n.
$$
When the sequence converges, both endpoints are its limit, proving \b[$\varphi(x)=\lim\alpha_n$]. These bounds also give $\|\varphi\|=1$ and positivity on nonnegative sequences, but do not by themselves assert shift invariance.