Subsequential limit

In mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In first-countable spaces, the two concepts coincide.

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. See limit superior and limit inferior.

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.