Divergent sequence

A sequence of points in a topological space without a limit. Every divergent sequence in a compact metric space contains a convergent subsequence. In the class of divergent sequences in a normed space one can find infinitely large sequences, i.e. sequences $\{x_n\}$ of points such that $\lim\limits_{n\to\infty}\|x_n\|=\infty$. The concept of a divergent sequence can be generalized to multiple sequences and to sequences in directed (partially ordered) sets.

See also: Convergence, types of.