A sequence in $X$ is a function $f: \mathbb{N}\to X$. # Definitions ###### In $\mathbb{R}^{n}$ A [[Real sequences|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 $|a_{n}-l|<\epsilon \quad \forall n \geq N.$ 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) $ If $(a_{n})$ converges to $l$ we write - $a_{n } \to l$ as $n \to \infty$ or $\lim_{ n \to \infty } a_{n} = l$. Convergence to infinity $a_{n} \to \infty$ as $n \to \infty$ for [[Properly Divergent Sequence]]. A [[Sequences|sequence]] $(x_{j})$ of vectors in $\mathbb{R}^n$ converges to $x\in \mathbb{R}^n$ iff $\forall \varepsilon > 0, \exists N \in \mathbb{N} \text{ such that }, \forall j\geq N, \lVert x_{j} - x \rVert <\varepsilon . $ ###### In normed spaces Let $(X, \lVert \cdot \rVert)$ be a [[Norms|normed space]]. We say that a sequence $(x_{n})_{n=1}^{\infty}$ in $X$ converges to $x\in X$ if and only if the sequence $\lVert x_{n} -x \rVert$ of real numbers converges to $0$, as $n\to \infty$. Since $0\leq\biggr\lvert \lVert x_{n} \rVert - \lVert x \rVert \biggr\rvert\leq \lVert x_{n} -x \rVert$, the sandwich rule yields that $\lVert x_{n} \rVert \to \lVert x \rVert$ as $n\to \infty$. ###### In metric spaces ###### In topological spaces Let $(X, \mathcal{T})$ be a [[Topology|topological space]] and $(x_{n})_{n=1}^\infty$ a [[Sequences|sequence]] in $X$. We say that $x_{n}\to x\in X$ as $n\to \infty$ only if for all $U \in \mathcal{T}$ containing $x$, there exists a natural numbers $N$, such that $x_{n}\in U$ for all $n \geq N$. # Properties # TBC This definition is equivalent to [[Real Cauchy Sequence]] by [[General Principle of Convergence]]. See [[Properties of Limits]]. See [[Examples of Convergent Sequences]]. See [[Tests for convergence of a sequence]]. See **applications** [[Convergent Real Series]] and [[Continuous Function (Sequential Continuity Definition)]].