I suppose you speak about Banach limits (as was suggested in the comments). You wrote that you only want a hint, I tried to give a complete answer. So if you want to try it by yourself, do not scroll completely to the end. I will assume that your definition of Banach limit is that it is positiv...

Suppose $(x_n)$ is bounded sequence. For any number $\theta\in [\liminf a_n,\limsup a_n]$, there exists a generalised limit functional that assigns that number $\theta$ as the "limit" of the sequence $(x_n)$. By Hahn-Banach theorem, we can extend this limit. This claim can be seen by firs...

Let $(\ell^{\infty})'$ be the $\mathbb{F}$-vector space of linear and continuous (bounded) functionals $\ell^{\infty}\rightarrow \mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$ (but we can assume $\mathbb{F}=\mathbb{R}$, if needed) and $\ell^{\infty}$ has the sup norm $\par...

Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm and let $c,c_0$ be the subspaces of sequences that are convergent, resp. convergent to zero. Show that: The linear functional $\ell_0\colon c\rightarrow \mathbb{C}$ defined for $x = (x_n) \in c$ by $$ \ell_0(x) = \li...

Let $\ell^{\infty}$ be the set of bounded sequences in $\mathbb{F}$, with the supremum norm. $c \subset \ell^{\infty}$ the sequences whose limit exists. Then there exists a $f \in (\ell^{\infty})'$, the dual, such that $f(x) = \lim_{n \to \infty} x(n)$ for all $x \in c$. Because we can define ...

What is a generalized limit?$\newcommand{\LIM}{\operatorname{LIM}}$ The question is about generalized limits $\LIM\colon\ell_\infty\to\mathbb R$. To deal with the question, we need to clearly define what we mean by a generalized limit. We want $\LIM$ to be a linear continuous functional on $\el...