> [!Definition] Definition (Convergence of sequence of real numbers) > A [[Real Sequence]] of real numbers $(a_{n})$ *converges* to a *limit* $l \in \mathbb{R}$ if for every $\epsilon>0$ there exists $N \in \mathbb{N}$ such that $|a_{n}-l|<\epsilon \quad \forall n \geq N.$ > >i.e. $\forall\epsilon>0 \;(\exists N\in\mathbb{N}\; (\forall n\geq N\; (|a_{n}-l|<\epsilon)))$ or $\forall\epsilon>0(\exists N\in\mathbb{N} \; (n\geq N \implies |a_{n}-l|<\epsilon))$. > > > >In other words, the terms of the sequence become arbitrarily close to L as n increases This definition is equivalent to [[Real Cauchy Sequence]] by [[General Principle of Convergence]]. **Negation** Negating gives a series does not converge to $l$ if $\exists\epsilon>0(\forall N\in \mathbb{N}(\exists n \geq N(|a_{n}-l|\geq \epsilon)))$. In other words there exists $\epsilon>0$ such that for all $N\in \mathbb{N}$ $\exists n \geq N \; (|a_{n}-l|\geq \epsilon) $**Notation** - If $(a_{n})$ converges to $l$ we write: - $a_{n } \to l$ as $n \to \infty$ or - $\lim_{ n \to \infty } a_{n} = l$. - $a_{n} \to \infty$ as $n \to \infty$ for [[Properly Divergent Sequence]]. See [[Properties of Limits]]. See [[Examples of Convergent Sequences]]. See [[Tests for convergence of a sequence]]. ### Remarks See **applications** [[Convergent Real Series]] and [[Continuous Function (Sequential Continuity Definition)]].