Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$

Question

Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$. For example, if I have a series $\{a_n\}_{n\in \mathbb{N}} \in [0,1]$, what does it mean that the limit of the series is $a(l)$ for $l \in \beta \mathbb{N} \setminus \mathbb{N}$? Why is this limit monotone and linear (it can't be just from definition can it?)

Answer 1

The linearity of the limit means the following: for every $l$ from βN∖N, and for every two sequences ($a_n$),($b_n$) and their continuous completion over βN ($A$),($B$)
B-lim (A+B)($l$) = B-lim A($l$) + B-lim B($l$)