limit inferior
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 inferior of , to be the infimum of all the limit points of . If there are no limit points, we define the limit inferior to be .
The two most common notations for the limit inferior are
and
An alternative, but equivalent, definition is available in the case of an infinite sequence of real numbers . For each , let be the infimum of the tail,
This construction produces a non-decreasing sequence
which either converges to its supremum, or diverges to . We define the limit inferior of the original sequence to be this limit;
Title | limit inferior |
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Canonical name | LimitInferior |
Date of creation | 2013-03-22 12:22:01 |
Last modified on | 2013-03-22 12:22:01 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 10 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 26A03 |
Synonym | liminf |
Synonym | infimum limit |
Related topic | LimitSuperior |