Can someone explain me what n, N represent and why N<n? Thank you.
If $\displaystyle (x_n)\to l$ means that almost all of the terms of $\displaystyle x(_n)$ are close to the number $\displaystyle l$.
Almost all means all but a finite collection of the terms.
If $\displaystyle N\in\mathbb{Z}^+$ then the set $\displaystyle \{1,2,\cdots N\}$ is a finite set and the set $\displaystyle \{n:n>N\}$ is an infinite set.
So in the that definition to say $\displaystyle n>N$ means that infinite set is almost all of the terms.