 Solution

A sublinear functional on a real vector space satisfies $p(x+y)\le p(x)+p(y)$ and $p(tx)=tp(x)$ for every $t\ge 0$. The dominated real Hahn-Banach theorem says that a linear functional on a subspace satisfying $f\le p$ extends to a linear functional on the whole space still dominated by $p$.

The limit-superior functional is finite on bounded sequences, positively homogeneous, and subadditive, since for every tail $\sup ({\alpha }_n+{\beta }_n)\le \sup {\alpha }_n+\sup {\beta }_n$. Passing to the tail limits proves sublinearity. On the subspace $c$ of convergent real sequences, the ordinary limit is linear and equals $p$. Hahn-Banach therefore extends it to $\phi$ on ${\ell }^{\infty }$ with