A sequence $x_0, x_1, x_2, \dots$ in a metric space $(X,d)$ is a convergent sequence if there exists a point $x \in X$ such that, for every real number $\epsilon > 0$ there exists a natural number $N$ such that $d(x,x_n) < \epsilon$ for all $n > N$

The point $x$ if it exists, is unique, and is called the limit point or limit of the sequence. One can also say that the sequence $x_0, x_1, x_2, \dots$ converges to $x$

A sequence is said to be divergent if it does not converge.