User:Caliburn/s/fa/Banach Limit Bounded Between Limit Inferior and Limit Superior

Theorem

Let $\map {\ell^\infty} \R$ be the vector space of bounded sequences.

Let $L$ be a Banach limit on $\map {\ell^\infty} \R$.

Then, for $x = \sequence {x_n}_{n \mathop \in \N} \in \map {\ell^\infty}\R$, we have:

$\ds \liminf_{n \mathop \to \infty} x_n \le \map L x \le \limsup_{n \mathop \to \infty} x_n$