Limes superior and limes inferior

Limes superior and limes inferior of a sequence: The sequence x _n is shown with blue dots. The two red curves approach the limes superior and limes inferior of the sequence, which are shown as dashed black lines.

In mathematics , Limes superior or Limes inferior of a sequence of real numbers denote the largest or smallest accumulation point of the sequence. Limes superior and limes inferior are partial replacements for the limit value if it does not exist.

notation

The limes inferior is referred to below with , the limes superior with . Common names are also used for the Limes inferior and for the Limes superior. ${\displaystyle \liminf _{n\rightarrow \infty }}$ ${\displaystyle \limsup _{n\rightarrow \infty }}$ ${\displaystyle \varliminf _{n\rightarrow \infty }}$ ${\displaystyle \varlimsup _{n\rightarrow \infty }}$

Limes superior and limes inferior for consequences

Sequences of real numbers

definition

Be a sequence of real numbers . Then the Limes inferior of is defined as ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

${\displaystyle \liminf _{n\rightarrow \infty }x_{n}:=\sup _{n\in \mathbb {N} }\,\inf _{k\geq n}x_{k}=\sup\{\inf\{x_{k}:k\geq n\}:n\in \mathbb {N} \}}$

Analogously, the Limes superior is defined as ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

${\displaystyle \limsup _{n\rightarrow \infty }x_{n}:=\inf _{n\in \mathbb {N} }\,\sup _{k\geq n}x_{k}=\inf\{\sup\{x_{k}:k\geq n\}:n\in \mathbb {N} \}.}$

And stand for Infimum and Supremum . ${\displaystyle \inf }$ ${\displaystyle \sup }$

properties

In the case of restricted sequences, almost all sequence members for each are in the open interval . ${\displaystyle \epsilon >0}$ ${\displaystyle (\liminf _{n\rightarrow \infty }x_{n}-\epsilon ,\limsup _{n\rightarrow \infty }x_{n}+\epsilon )}$

Limes inferior and limes superior exist as elements of the extended real numbers for every sequence of real numbers. The limes inferior and the limes superior are both real numbers if and only if the sequence is bounded. In this case, from the existence of Limes inferior and Limes superior, one obtains the Bolzano-Weierstrass theorem . ${\displaystyle \mathbb {R} \cup \lbrace -\infty ,+\infty \rbrace }$

For each there are infinitely many terms in the open interval ${\displaystyle \epsilon >0}$

${\displaystyle (\limsup _{n\rightarrow \infty }x_{n}-\epsilon ,\limsup _{n\rightarrow \infty }x_{n}+\epsilon )}$ or. ${\displaystyle (\liminf _{n\rightarrow \infty }x_{n}-\epsilon ,\liminf _{n\rightarrow \infty }x_{n}+\epsilon ).}$

In addition, almost all of the elements in the sequence fulfill

${\displaystyle \liminf _{n\rightarrow \infty }x_{n}-\epsilon <x_{n}<\limsup _{n\rightarrow \infty }x_{n}+\epsilon .}$

The limes inferior is the smallest and the limes superior is the largest accumulation point of a sequence and thus applies

${\displaystyle \liminf _{n\rightarrow \infty }x_{n}\leq \limsup _{n\rightarrow \infty }x_{n}.}$

Equality is present if and only if the sequence converges in the expanded real numbers . In this case

${\displaystyle \lim _{n\rightarrow \infty }x_{n}=\liminf _{n\rightarrow \infty }x_{n}=\limsup _{n\rightarrow \infty }x_{n}.}$

The designation or is motivated by the fact that ${\displaystyle \limsup }$ ${\displaystyle \liminf }$

${\displaystyle \liminf _{n\rightarrow \infty }x_{n}=\lim _{n\rightarrow \infty }\left(\inf _{k\geq n}x_{k}\right)}$ or. ${\displaystyle \limsup _{n\rightarrow \infty }x_{n}=\lim _{n\rightarrow \infty }\left(\sup _{k\geq n}x_{k}\right).}$

The limit values ​​exist because monotonic sequences are convergent in the expanded real numbers.

Since accumulation points are precisely the limit values ​​of convergent subsequences, the inferior limes is the smallest extended real number to which a subsequence converges or the superior limes is the largest.

Generalization to general consequences

Let be a partially ordered set and a sequence. In order to be able to define and just as in the case of real sequences, the corresponding suprema and infima must exist. This is the case, for example, when there is a complete lattice , so that in this case too, each sequence has an inferior limes and a superior limes. ${\displaystyle M}$ ${\displaystyle f\colon \mathbb {N} \to M}$ ${\displaystyle \liminf _{n\rightarrow \infty }x_{n}}$ ${\displaystyle \limsup _{n\rightarrow \infty }x_{n}}$ ${\displaystyle M}$ ${\displaystyle M}$

Limes superior and Limes inferior for sequences of real functions

For a sequence of real functions , i.e. for all , limes inferior and limes superior are defined point by point, i.e. ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle (\liminf _{n\rightarrow \infty }f_{n})(x)=\liminf _{n\to \infty }f_{n}(x){\text{ und }}(\limsup _{n\rightarrow \infty }f_{n})(x)=\limsup _{n\to \infty }f_{n}(x).}$

A well-known mathematical statement that uses the term Limes inferior of a sequence of functions is Fatou's lemma .

Limes superior and limes inferior of sequence of sets

→ Main article : Limes superior and Limes inferior of set sequences

For any set , the power set forms a complete lattice under the order defined by the subset relation. Let be a sequence of subsets of , i.e. for all . Then applies ${\displaystyle \Omega }$ ${\displaystyle P(\Omega )}$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle A_{n}\subseteq \Omega }$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \sup _{n\in \mathbb {N} }A_{n}=\bigcup _{n\in \mathbb {N} }A_{n},\inf _{n\in \mathbb {N} }A_{n}=\bigcap _{n\in \mathbb {N} }A_{n}.}$

This gives for Limes inferior and Limes superior

${\displaystyle \liminf _{n\rightarrow \infty }A_{n}={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}A_{m}\right)}$

and

${\displaystyle \limsup _{n\rightarrow \infty }A_{n}={\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}A_{m}\right).}$

The limes inferior of a sequence can be described as the set of all elements from that lie in almost all , the limes superior to the set sequence as the set of all elements from that lie in an infinite number . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle A_{n}}$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle A_{n}}$

For example, the superior limes of sets is used in the Borel-Cantelli lemma . In addition, convergent set sequences can be defined with the Limes inferior and superior . The sequence is said to converge to a set if the inferior limes and superior limes of the sequence are equal. A sequence of subsets of a set converges if and only if there is an index for each such that either applies to all or to all . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle N=N(x)}$ ${\displaystyle x\in A_{n}}$ ${\displaystyle n\geq N}$ ${\displaystyle x\notin A_{n}}$ ${\displaystyle n\geq N}$

Limes superior and limes inferior of functions

Let be an interval , an interior point of, and a real-valued function. Then Limes superior and Limes inferior are those values ​​from the extended real numbers that are defined as follows: ${\displaystyle I\subseteq \mathbb {R} }$ ${\displaystyle \xi }$ ${\displaystyle I}$ ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle \mathbb {R} \cup \{-\infty ,+\infty \}}$

${\displaystyle \limsup _{x\to \xi }f(x)=\inf _{a>0}\sup f((\xi -a,\xi +a)\backslash \{\xi \})}$,

${\displaystyle \liminf _{x\to \xi }f(x)=\sup _{a>0}\inf f((\xi -a,\xi +a)\backslash \{\xi \})}$.

${\displaystyle f((\xi -a,\xi +a))}$denotes the image set of the open interval , choosing so small that . ${\displaystyle (\xi -a,\xi +a)}$ ${\displaystyle a}$ ${\displaystyle (\xi -a,\xi +a)\subseteq I}$

Analogous to unilateral limit values, a unilateral limes superior and a unilateral limes inferior are defined:

${\displaystyle \limsup _{x\to \xi +}f(x)=\inf _{a>0}\sup f((\xi ,\xi +a))}$,

${\displaystyle \liminf _{x\to \xi +}f(x)=\sup _{a>0}\inf f((\xi ,\xi +a))}$,

${\displaystyle \limsup _{x\to \xi -}f(x)=\inf _{a>0}\sup f((\xi -a,\xi ))}$,

${\displaystyle \liminf _{x\to \xi -}f(x)=\sup _{a>0}\inf f((\xi -a,\xi ))}$.

Limes superior and limes inferior of functions are used, for example, in the definition of semi-continuity .

Generalization of Limes superior and Limes inferior

definition

Let be an arbitrary topological space, a partially ordered set, in which for every nonempty subset both and exist. bear the topology induced by this order. Be wider , and be a point of accumulation of (that is, every neighborhood of contains an element from different from ). The amount of the environments in going with designated. ${\displaystyle T}$ ${\displaystyle M}$ ${\displaystyle A\subseteq M}$ ${\displaystyle \inf A}$ ${\displaystyle \sup A}$ ${\displaystyle M}$ ${\displaystyle f:V\rightarrow M}$ ${\displaystyle V\subseteq T}$ ${\displaystyle a\in T}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle V}$ ${\displaystyle {\mathfrak {U}}(a)}$

Now define:

${\displaystyle \limsup _{x\to a}f(x):=\inf _{U\in {\mathfrak {U}}(a)}\sup _{x\in U\backslash \{a\}}f(x)}$

${\displaystyle \liminf _{x\to a}f(x):=\sup _{U\in {\mathfrak {U}}(a)}\inf _{x\in U\backslash \{a\}}f(x)}$

${\displaystyle {\mathfrak {U}}(a)}$may be replaced by any environment base of . ${\displaystyle a}$

properties

It is always

${\displaystyle \liminf _{x\to a}f(x)\leq \limsup _{x\to a}f(x)}$

In addition, it follows from the equality of the limes superior and the limes inferior that it exists and it holds ${\displaystyle \liminf _{x\to a}f(x)=\limsup _{x\to a}f(x)}$ ${\displaystyle \lim _{x\to a}f(x)}$

${\displaystyle \lim _{x\to a}f(x)=\liminf _{x\to a}f(x)=\limsup _{x\to a}f(x)}$

Examples

• For , , and one obtains the known from calculus definition of Limes inferior and superior limit of a sequence of real numbers. ${\displaystyle T=\mathbb {N} \cup \{\infty \}}$ ${\displaystyle V=\mathbb {N} }$ ${\displaystyle M=\mathbb {R} \cup \{-\infty ,\infty \}}$ ${\displaystyle a=\infty }$
• For , , and gives the definition of Limes inferior and superior limit for quantity consequences. ${\displaystyle T=\mathbb {N} \cup \{\infty \}}$ ${\displaystyle V=\mathbb {N} }$ ${\displaystyle M=\operatorname {Pot} (\Omega )}$ ${\displaystyle a=\infty }$

literature

• Konrad Königsberger : Analysis 1st 6th edition, Springer, Berlin / Heidelberg / New York 2004, ISBN 3-540-41282-4 , p. 50.
• Heinz Bauer : Measure and integration theory. 2nd Edition. De Gruyter, Berlin 1992, ISBN 3-11-013626-0 (hardcover), ISBN 3-11-013625-2 (paperback), p. 93 (on sequences of quantities).

Individual evidence

1. ↑ Nelson Dunford and Jacob T. Schwartz. Linear operators. Part I. General Theory. John Wiles and Sons, 1988, p. 4. ISBN 0-471-60848-3 .