A sequence x0,x1,x2,… in a metric space (X,d) is a convergent sequence if there exists a point x∈X such that, for every real number ϵ>0, there exists a natural number N such that d⁢(x,xn)<ϵ 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 x0,x1,x2,… converges to x.
A sequence is said to be divergent if it does not converge.