\begin{definition}[Definition:Almost Convergent Sequence]
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 $L$ {{iff}}:
:$\map \phi {\sequence {x_n}_{n \mathop \in \N} } = L$
for each Banach limit $\phi$.
\end{definition}