Limit of a sequence (${\lim }_{n\to \infty }{x}_n$)

A real sequence $(x_n)$ has limit of a sequence $L\in \mathbb{R}$ when for every $\epsilon >0$ there is $N$ such that $|x_n-L|<\epsilon$ for all $n\ge N$. Its limit of a sequence is unique. Extended-real convergence to $-\infty$ means that every real upper bound eventually exceeds all the terms; convergence to $+\infty$ is the corresponding lower-bound condition. The extended-real Fekete lemma is an example where the limit of a sequence need not be finite.