# Subsequential limit

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{{Short description|Limit of some subsequence}}
{{refimprove|date=April 2023}}
In [mathematics](/source/mathematics), a '''subsequential limit''' of a [sequence](/source/sequence) is the [limit](/source/Limit_of_a_sequence) of some [subsequence](/source/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](/source/cluster_point), but not conversely. In [first-countable](/source/First-countable_space) 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 [almost everywhere convergence](/source/Pointwise_convergence).

The [supremum](/source/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](/source/limit_superior_and_limit_inferior).<ref name="ross" />

If <math>(X, d)</math> is a [metric space](/source/metric_space) and there is a [Cauchy sequence](/source/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

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Adapted from the Wikipedia article [Subsequential limit](https://en.wikipedia.org/wiki/Subsequential_limit) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Subsequential_limit?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.