Definition:Limit Superior

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Definition

Let $\sequence {x_n}$ be a bounded sequence in $\R$.


Definition 1

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$.


From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum.

This maximum is called the limit superior.

It can be denoted:

$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$


Definition 2

The limit superior of $\sequence {x_n}$ is defined and denoted as:

$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \inf \set {\sup_{m \mathop \ge n} x_m: n \in \N}$


Also known as

The limit superior is also called the upper limit, or just limsup.

Some sources use the term upper bound, but that term has a wider application and is not recommended in this context.


Examples

Sequence of Reciprocals

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \dfrac 1 n$


The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 0$


Divergent Sequence $\paren {-1}^n$

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \paren {-1}^n$


The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$


Farey Sequence

Consider the Farey sequence:

$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$


The limit superior of $\sequence {a_n}$ is given by:

$\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$


Also see

  • Results about limits superior can be found here.


Linguistic Note

The plural of limit superior is limits superior.

This is because limit is the noun and superior is the adjective qualifying that noun.


Sources

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