Limit of a sequence
= Limit of a sequence
{title2=$\lim_{n\to\infty}x_n$}
A real <sequence> $(x_n)$ has <limit of a sequence> $L\in\mathbb R$ when for every $\varepsilon>0$ there is $N$ such that $|x_n-L|<\varepsilon$ for all $n\geq N$. 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.