limit superior
Let be a set of real numbers. Recall that a limit
point of is a real number such that for all
there exist infinitely many such that
We define , pronounced the
limit superior of , to be the supremum of all the limit
points of . If there are no limit points, we define the limit
superior to be .
We can generalize the above definition to the case of a mapping . Now, we define a limit point of to be an such that for all there exist infinitely many such that
We then define , to be the supremum of all the limit points of , or if there are no limit points. We recover the previous definition as a special case by considering the limit superior of the inclusion mapping .
Since a sequence of real numbers is just a
mapping from to , we may adapt the above definition
to arrive at the notion of the limit superior of a sequence. However
for the case of sequences, an alternative, but equivalent definition
is available. For each , let be the supremum of
the tail,
This construction produces a non-increasing sequence
which either converges to its infimum, or diverges to .
We define the limit superior of the original sequence to be this limit;
Title | limit superior |
---|---|
Canonical name | LimitSuperior |
Date of creation | 2013-03-22 12:21:58 |
Last modified on | 2013-03-22 12:21:58 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 12 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 26A03 |
Synonym | limsup |
Synonym | supremum limit |
Related topic | LimitInferior |