Limit of a sequence of real numbers

Notation: 
* a_n \to L as n \to \infty
* \lim_{n \to \infty} a_n = L
== Definition ==
Let (a_n)_n = a_1, a_2, ... be a sequence of real numbers. If the limit of this sequence exists, L \in \mathbb{R} is the limit of the sequence if and only if: for every \varepsilon > 0, \exists N_\varepsilon such that \forall n \geq N_\varepsilon, we have| a_n - L | < \varepsilon.