Here is a problem from an exam I just took several days ago.
Let $\ell^\infty$ be the real Banach space consists of real sequences $a=(a_1,a_2, \cdots )$ for which $\sup_{n\in\mathbb N}|a_n|<\infty$ and equipped with the norm $\|a\|=\sup_{n\in\mathbb N}|a_n|$. Show that there is a continuous linear functional $f\in(\ell^\infty)^*$ such that for all $a=(a_1,a_2, \cdots )\in\ell^\infty$, we have:
$\liminf_{n\to\infty}(a_n+a_{n+1})\leq f(a)\leq \limsup_{n\to\infty}(a_n+a_{n+1})$ and
$f(a_1,a_2,\cdots)=f(a_2,a_3,\cdots)$.
Let $$X_0=\{a=(a_1,a_2, \cdots )\in\ell^\infty: \lim_{n\to\infty}a_n \text{ exists}\}.$$ Then $X_0$ is a sub linear space of $\ell^\infty$. Define $$p(a)=\limsup_{n\to\infty}(a_n+a_{n+1}),\ \ \ \forall a=(a_1,a_2, \cdots )\in \ell^\infty.$$ It is easy to check that $p$ is a sub-linear functional on $\ell^\infty$. Define $$f_0(a)=2\lim_{n\to\infty}a_n,\ \ \ \forall a=(a_1,a_2, \cdots )\in X_0.$$ Then $f_0$ is a linear functional on $X_0$ with $f_0=p|_{X_0}$. By Hahn-Banach theorem, there is a linear functional $f$ on $\ell^\infty$ that extends $f_0$ and satisfies $$f(a)\leq \limsup_{n\to\infty}(a_n+a_{n+1})\ \ \ \forall a=(a_1,a_2, \cdots )\in \ell^\infty.$$ Now it is not hard to chaeck that $f$ is continuous and satisfies the first requirement.
But I don't know how to show that this $f$ also satisfies the second requirement. It is clear that it has the second property on $X_0$. Does it imply that the second condition also holds on the whole $\ell^\infty$? Or we need to choose another $f$ with refiner requirements?
Any help would be appreciated.
