Subsequential limit

(UTC)In mathematics, a subsequential limit is the limit of some subsequence.

The supremum of the set of all subsequential limits of some sequence is called the limit superior, or limsup. Similarly, the infimum of such a set is called the limit inferior, or liminf.

It can be shown that if (X,d) is a metric space and there is a Cauchy sequence such that there is a subsequence converging to some x, then the sequence also converges to x.