In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a subsequential limit of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
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 is a ]metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
and there is a Cauchy sequence such that there is a subsequence converging to some then the sequence also converges to
See also
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References
{{Mathanalysis-stub
Limits (mathematics)
Sequences and series