Convergence

A sequence $x_0, x_1, x_2, \dots$ in a metric space $(X,d)$ is called a convergent if there exists a point $a \in X$ such that, for every real number $\epsilon > 0$, there exists a natural number $N$ such that $d(a,x_n) < \epsilon$ for all $n > N$. If such a number doesn't exist, the sequence is called divergent.

The point $a$, if it exists, is unique. Then we say that this point is the limit of the sequence and that the sequence $x_0, x_1, x_2, \dots$ converges to $a$.