Limit of a Sequence of Real Numbers

Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers, and let $L\in\mathbb{R}$. One says that $(a_n)$ converges to $L$ if for every $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that for all integers $n\ge N$,

$$|a_n-L|<\varepsilon.$$

In that case one writes $\lim_{n\to\infty} a_n = L$ or $a_n\to L$ as $n\to\infty$.