Editing Subsequential limit

{{Short description|The limit of some subsequence}}
{{refimprove|date=April 2023}}
In [[mathematics]], a '''subsequential limit''' of a [[sequence]] is the [[Limit of a sequence|limit]] of some [[subsequence]].<ref name="ross">{{cite book |last1=Ross |first1=Kenneth A. |title=Elementary Analysis: The Theory of Calculus |date=3 March 1980 |publisher=Springer |isbn=9780387904597 |url=https://books.google.com/books?id=5JxHZNpMq3AC |access-date=5 April 2023}}</ref> Every subsequential limit is a [[cluster point]], but not conversely. In [[First-countable space|first-countable]] spaces, the two concepts coincide.

In a topological space, if every subsequence has a subsequential limit to the same point, then the original sequence also converges to that limit. This need not hold in more generalized notions of convergence, such as the space of [[Pointwise convergence#Almost everywhere convergence|almost everywhere convergence]].

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]].<ref name="ross" />

If <math>(X, d)</math> is a [[metric space]] and there is a [[Cauchy sequence]] such that there is a subsequence converging to some <math>x,</math> then the sequence also converges to <math>x.</math>

==See also==

* {{annotated link|Convergent filter}}
* {{annotated link|List of limits}}
* {{annotated link|Limit of a sequence}}
* {{annotated link|Limit superior and limit inferior}}
* {{annotated link|Net (mathematics)}}
* {{annotated link|Filters in topology#Subordination analogs of results involving subsequences}}

==References==
{{reflist}}

{{Topology}}
{{Mathanalysis-stub}}

[[Category:Limits (mathematics)]]
[[Category:Sequences and series]]