Convergent sequence

A sequence whose terms eventually become arbitrarily close to a single limit point.

Convergent sequence

Let $(X,d)$ be a metric space and let $(x_n)_{n\in\mathbb{N}}$ be a sequence in $X$ . The sequence is convergent if there exists $x\in X$ such that

\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n\ge N,\ d(x_n,x)<\varepsilon.

In that case, one writes $x_n\to x$ and calls $x$ the limit .

Convergence is the basic notion underlying limits of functions , continuity , series , and completeness . In a metric space, limits (if they exist) are unique.

Examples:

In $\mathbb{R}$ , $x_n=1/n$ converges to $0$ .
In $\mathbb{R}^2$ , $x_n=(1/n,(-1)^n/n)$ converges to $(0,0)$ .
In a discrete metric space, a sequence converges iff it is eventually constant.