Definition:Almost Convergent Sequence

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

Let $\sequence {x_n}_{n \mathop \in \N} \in \map {\ell^\infty} \R$.

We say that $\sequence {x_n}_{n \mathop \in \N}$ is almost convergent to $L$ :


 * $\map \phi {\sequence {x_n}_{n \mathop \in \N} } = L$

for each Banach limit $\phi$.