Misplaced Pages

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Subsequential limit" – news · newspapers · books · scholar · JSTOR (April 2023) (Learn how and when to remove this message)

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.

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.

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.

If ${\displaystyle (X,d)}$ is a metric space and there is a Cauchy sequence such that there is a subsequence converging to some ${\displaystyle x,}$ then the sequence also converges to ${\displaystyle x.}$

See also

• Convergent filter – Use of filters to describe and characterize all basic topological notions and results.Pages displaying short descriptions of redirect targets
• List of limits
• Limit of a sequence – Value to which tends an infinite sequence
• Limit superior and limit inferior – Bounds of a sequencePages displaying short descriptions of redirect targets
• Net (mathematics) – A generalization of a sequence of points
• Filters in topology#Subordination analogs of results involving subsequences – Use of filters to describe and characterize all basic topological notions and results.

References

1. ^ Ross, Kenneth A. (3 March 1980). Elementary Analysis: The Theory of Calculus. Springer. ISBN 9780387904597. Retrieved 5 April 2023.

v t e Topology
Fields	General (point-set) Algebraic Combinatorial Continuum Differential Geometric low-dimensional Homology cohomology Set-theoretic Digital
Key concepts	Open set / Closed set Interior Continuity Space compact connected Hausdorff metric uniform Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold Bundle (mathematics) Second-countable space Cobordism
Metrics and properties	Euler characteristic Betti number Winding number Chern number Orientability
Key results	Banach fixed-point theorem De Rham cohomology Invariance of domain Poincaré conjecture Tychonoff's theorem Urysohn's lemma
Category Mathematics portal Wikibook Wikiversity Topics general algebraic geometric Publications

This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Mathematical analysis stubs
• Limits (mathematics)
• Sequences and series