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The upper (lower) limit of a sequence is the largest (respectively, smallest) [[Limit|limit]] among all partial (finite and infinite) limits of a given sequence of real numbers. For any sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958302.png" /> the set of all its partial limits (finite and infinite) on the extended number axis (i.e. in the set of real numbers, completed by the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958304.png" />) is non-empty and has both a largest and a smallest element (finite or infinite). The largest element of the set of partial limits is said to be the upper limit (lim sup) of the sequence and is denoted by
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958305.png" /></td> </tr></table>
+
==Upper and lower limit of a real sequence==
 +
===Definition===
 +
The upper and lower limit of a sequence of real numbers $\{x_n\}$ (called also limes superior and limes inferior) can be defined in several ways and are denoted, respectively as
 +
\[
 +
\limsup_{n\to\infty}\, x_n\qquad \liminf_{n\to\infty}\,\, x_n
 +
\]
 +
(some authors use also the notation $\overline{\lim}$ and $\underline{\lim}$). One possible definition is the following
  
while the smallest element is said to be the lower limit (lim inf) and is denoted by
+
'''Definition 1'''
 +
\[
 +
\limsup_{n\to\infty} \, x_n = \inf_{n\in\mathbb N}\,\,  \sup_{k\geq n}\, x_k
 +
\]
 +
\[
 +
\liminf_{n\to\infty}\,<, x_n = \sup_{n\in\mathbb N}\,\, \inf_{k\geq n}\, x_k\, .
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958306.png" /></td> </tr></table>
+
===Properties===
 +
It follows easily from the definition that
 +
\[
 +
\liminf_n\,\, x_n = -\limsup_n\, (-x_n)\, ,
 +
\]
 +
\[
 +
\liminf_n\,\, (\lambda x_n) = \lambda\, \liminf_n\,\, x_n\qquad \limsup_n\, (\lambda x_n) = \lambda\, \limsup_n\, x_n\qquad \mbox{when $\lambda > 0$}
 +
\]
 +
and that
 +
\[
 +
\liminf_n\,\, (x_n + y_n)\geq \liminf\, x_n + \liminf\,\, y_n \qquad \limsup_n\, (x_n + y_n)\leq \limsup\, x_n + \limsup\, y_n
 +
\]
 +
if the additions are not of the type $-\infty + \infty$.
  
For instance, if
+
Moreover, the limit of $\{x_n\}$ exists and it is a real number $L$ (respetively $\infty$, $-infty$) if and only if the upper and lower limit coincide
 +
and are a real number $L$ (resp. $\infty$, $-\infty$).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958307.png" /></td> </tr></table>
+
The upper and lower limits of a sequence are both finite if and only if the sequence is bounded.
  
then
+
===Characterizations===
 +
The upper and lower limits can also be defined in several alternative ways. In particular
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958308.png" /></td> </tr></table>
+
'''Theorem 1'''
 +
Let $S:=\{a\in ]-\infty, \infty] : \{k: x_k >a\} \mbox{is finite}\}$ and $L:= \{a\in  [-\infty, \infty[ : \{k: x_k <a\} \mbox{is finite}\}$. Then
 +
$\limsup x_n$ is the minimum of $S$ and $\liminf x_n$ is the maximum of $L$.
  
If
+
'''Theorem 2'''
 +
Consider the set $A$ of elements $\ell\in [-\infty, \infty]$ for which there is a subsequence of $\{y_n\}$ converging to $\ell$. Then
 +
$\limsup x_n$ is the maximum of $A$ and $\liminf x_n$ is the minimum of $A$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u0958309.png" /></td> </tr></table>
+
'''Theorem 3'''
 +
$U:=\limsup x_n$ is characterized by the two properties:  
 +
* if $U< \infty$ for all $u> U$ there is $N\in \mathbb N$ such that $x_n< u$ for all $n> N$;
 +
* if $U> -\infty$ for all $u< U$ and $N\in \mathbb N$ there is a $k>N$ with $x_k> u$.
 +
$L:=\liminf x_n$ is characterized by the two properties:
 +
* if $U> -\infty$ for all $u< U$ there is $N\in\mathbb N$ such that $x_n> u$ for all $n> N$;
 +
* if $U< \infty$ for all $u> U$ and $N\in\mathbb N$ there is a $k> N$ with $x_k< u$.
  
then
+
===Examples===
 +
If $x_n = (-1)^n$ then
 +
\[
 +
\liminf\, x_n = -1 \qquad \mbox{and} \qquad \limsup_n\, x_n = 1\, .
 +
\]
 +
If $x_n = (-1)^n n$ then
 +
\[
 +
\liminf\, x_n = -\infty \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, .
 +
\]
 +
If $x_n = n + (-1)^n$, then
 +
\[
 +
\liminf\, x_n = 0 \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, .
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583010.png" /></td> </tr></table>
+
==Upper and lower limit of a real function==
 +
===Definition===
 +
If $f$ is a real-valued function defined on a set $E\subset \mathbb R$ (or $\subset \mathbb R^k$), the upper and lower limits of $f$ at $x_0$ are denoted by
 +
\[
 +
\limsup_{x\to x_0}\, f(x)\qquad \mbox{and}\qquad \liminf_{x\to x_0}\, f(x)\, .
 +
\]
 +
and, under the assumptions that $x_0$ is an [[Accumulation point|accumulation point]] for $E$ (i.e. there is a sequence $\{y_n\}\subset E\setminus x_0\}$ converging to $x_0$) can be defined as
  
If
+
'''Definition 4'''
 +
\[
 +
\limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\}
 +
\]
 +
\[
 +
\liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\}
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583011.png" /></td> </tr></table>
+
(Some authors include also the point $x_0$ in the definitions above, however this choice is less common).
  
then
+
The definition above can be easilily extended to functions defined on an arbitrary [[Metric space|metric space]] $(X, d)$: it suffices to replace
 +
$|x-x_0|< r$ with $d (x, x_0)< r$, namely
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583012.png" /></td> </tr></table>
+
'''Definition 5'''
 +
\[
 +
\limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\}
 +
\]
 +
\[
 +
\liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\}
 +
\]
  
Any sequence has a lim sup and a lim inf, and if the sequence is bounded from above (from below) its lim sup (lim inf) is finite. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583013.png" /> is the lim sup (lim inf) of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583015.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583016.png" /> the following conditions are fulfilled: a) there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583017.png" /> such that for all indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583018.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583020.png" />) is true; b) for any index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583021.png" /> there exists an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583025.png" />). The meaning of condition a) is that for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583026.png" /> there exists in the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583027.png" /> only a finite number of terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583030.png" />). The meaning of condition b) is that there exists an infinite set of terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583032.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583033.png" />). Provided both are finite, the lim inf can be reduced to the lim sup by changing the signs of the terms of the sequence:
+
As in the case of sequences, some authors use the notation $\overline{\lim}$ and $\underline{\lim}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583034.png" /></td> </tr></table>
+
===Characterizations===
 +
The upper and lower limits
 +
\[
 +
U =\limsup_{x\to x_0}\, f(x)
 +
\]
 +
\[
 +
L :=\liminf_{x\to x_0}\, f(x)
 +
\]
 +
can also be defined in several alternative ways. A useful one, which reduces to sequences, is the following:
  
For a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583036.png" /> to have a limit (finite or infinite (equal to one of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583037.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583038.png" />)) it is necessary and sufficient that
+
'''Theorem 6'''
 +
$U$ is characterized by the properties:
 +
* There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = U$;
 +
* For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\limsup\, f (y_k)\leq U$.
 +
$L$ is characterized by the properties:
 +
* There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = L$;
 +
* For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\liminf\, f (y_k)\geq L$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583039.png" /></td> </tr></table>
+
This theorem is valid in an arbitrary metric space.
  
The upper (lower) limit of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583040.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583041.png" /> is the limit of the upper (lower) bounds of the sets of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583042.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583043.png" />, when this neighbourhood contracts towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583044.png" />. These limits are denoted by
+
===Properties===
 +
From Theorem 6 it can be easily concluded that
 +
\[
 +
\liminf_{x\to x_0}\,\, f(x) = - \limsup_{x\to x_0}\, (- f(x))
 +
\]
 +
\[
 +
\liminf_{x\to x_0}\,\, (\lambda f (x)) = \lambda\,  \liminf_{x\to x_0}\,\, f(x) \qquad \limsup_{x\to x_0}\, (\lambda f(x)) = \lambda\, \limsup_{x\to x_0}\, f(x)\qquad \mbox{when $\lambda > 0$}
 +
\]
 +
and that
 +
\[
 +
\liminf_{x\to x_0}\,\, (f(x) + g(x))\geq \liminf_{x\to x_0}\,\, f(x) + \liminf_{x\to x_0}\,\, g(x) \qquad \limsup_{x\to x_0}\, (f(x) + g(x))\leq \limsup_{x\to x_0}\, f(x) + \limsup_{x\to x_0}\, g(x)
 +
\]
 +
if the additions are not of the type $-\infty + \infty$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583045.png" /></td> </tr></table>
+
The function $f$ has a finite limit $L$ (resp. has limit $\infty$, $-\infty$) if the lower and upper limits coincide and are equal to $L$ (resp. $\infty$, $-\infty$).
  
Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583046.png" /> be defined on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583047.png" /> and assume real values. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583049.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583050.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583052.png" />, then
+
Moreover, we have the following
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583053.png" /></td> </tr></table>
+
'''Proposition'''
 +
Consider the closed set $E'$ of accumulation points of $E$ and define
 +
\[
 +
\underline{f} (x) := \liminf_{y\to x}\,\, f(y)\,
 +
\]
 +
\[
 +
\overline{f} (x) := \limsup_{y\to x}\, f(y)\, .
 +
\]
 +
$\underline{f}$ is lower [[Semi-continuous function|semicontinuous]] and $\overline{f}$ is upper semicontinuous.
  
and
+
===From metric spaces to sequences===
 +
Consider the space $X=\mathbb N\cup\{\infty\}$ with the metric:
 +
\[
 +
d (n,m) = \left|\frac{1}{m+1} - \frac{1}{n+1}\right|,\qquad d(n,\infty) = \frac{1}{n+1}\, .
 +
\]
 +
Given a sequence of real numbers $\{x_n\}$ consider the function $f:\mathbb N\to \mathbb R$ given by $f(n) = x_n$ as a function defined on a subset of $X$. Then $\limsup_n\, x_n$ and $\liminf_n\,\, x_n$ in the sense of Definition 1 coincide with
 +
\[
 +
\limsup_{n\to\infty}\, f(n)\qquad \liminf_{n\to\infty}\,\, f (n)
 +
\]
 +
in the metric-space sense of Definition 6.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583054.png" /></td> </tr></table>
+
==Upper and lower limit of sets==
 +
If $\{A_k\}$ is a sequence of subsets of $X$, the upper and lower limit of the sequence $\{A_k\}$ is defined as
 +
\[
 +
\limsup_{k\to\infty}\, A_k = \bigcap_{n\in\mathbb N}\, \bigcup_{k\geq n} A_k\,
 +
\]
 +
\[
 +
\liminf_{k\to\infty}\, A_k = \bigcup_{n\in\mathbb N}\, \bigcap_{k\geq n} A_k\,.
 +
\]
  
At each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583055.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583056.png" /> has both an upper limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583057.png" /> and a lower limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583058.png" /> (finite or infinite). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583059.png" /> is upper semi-continuous, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583060.png" /> is lower semi-continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583061.png" /> (in the sense of the concept of semi-continuity of functions which assume values in the extended number axis, cf. also [[Semi-continuous function|Semi-continuous function]]).
+
==References==
 
+
{|
For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583062.png" /> to have a finite or infinite (equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583063.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583064.png" />) limit at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583065.png" /> it is necessary and sufficient that
+
|-
 
+
|valign="top"|{{Ref|Ap}}||valign="top"|  T.M. Apostol,   "Mathematical analysis". Second edition. Addison-Wesley  (1974) {{MR|0344384}} {{ZBL|0309.2600}}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583066.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|IlPo}}||valign="top"| V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian) {{MR|0687827}}   {{ZBL|0138.2730}}
The concept of the upper limit (lower limit) of a function at a point can be naturally extended to real-valued functions defined on topological spaces.
+
|-
 
+
|valign="top"|{{Ref|Ku}}||valign="top"| L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow  (1973)   (In Russian) {{MR|0619214}} {{ZBL|0703.26001}}
The upper and lower limit of a sequence of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583068.png" /> are the set
+
|-
 
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|valign="top"|{{Ref|Ni}}||valign="top"| S.M. Nikol'skii,  "A course of mathematical analysis" , '''1''' , MIR   (1977)  (Translated from Russian) {{MR|0466435}} {{ZBL|0384.00004}}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583069.png" /></td> </tr></table>
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|valign="top"|{{Ref|Ru}}||valign="top"| W. Rudin, "Principles of mathematical analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}} {{ZBL|0346.2600}} 
consisting of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583070.png" /> which belong to an infinite number of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583071.png" />, and the set
+
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+
|}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583072.png" /></td> </tr></table>
 
 
 
of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583073.png" /> which belong to all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583074.png" />, starting from some index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583075.png" />, respectively. Obviously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583076.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.D. Kudryavtsev,  "A course of mathematical analysis" , '''1''' , Moscow  (1988) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
The upper limit is also called the limes superior or limit superior, and the lower limit the limes inferior or limit inferior. Cf. also [[Upper and lower bounds|Upper and lower bounds]].
 
 
 
The limit superior and limit inferior of a sequence of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583077.png" /> of a set are given by the formulas
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583078.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095830/u09583079.png" /></td> </tr></table>
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill (1953)</TD></TR></table>
 

Revision as of 15:58, 12 August 2012


Upper and lower limit of a real sequence

Definition

The upper and lower limit of a sequence of real numbers $\{x_n\}$ (called also limes superior and limes inferior) can be defined in several ways and are denoted, respectively as \[ \limsup_{n\to\infty}\, x_n\qquad \liminf_{n\to\infty}\,\, x_n \] (some authors use also the notation $\overline{\lim}$ and $\underline{\lim}$). One possible definition is the following

Definition 1 \[ \limsup_{n\to\infty} \, x_n = \inf_{n\in\mathbb N}\,\, \sup_{k\geq n}\, x_k \] \[ \liminf_{n\to\infty}\,<, x_n = \sup_{n\in\mathbb N}\,\, \inf_{k\geq n}\, x_k\, . \]

Properties

It follows easily from the definition that \[ \liminf_n\,\, x_n = -\limsup_n\, (-x_n)\, , \] \[ \liminf_n\,\, (\lambda x_n) = \lambda\, \liminf_n\,\, x_n\qquad \limsup_n\, (\lambda x_n) = \lambda\, \limsup_n\, x_n\qquad \mbox{when '"`UNIQ-MathJax4-QINU`"'} \] and that \[ \liminf_n\,\, (x_n + y_n)\geq \liminf\, x_n + \liminf\,\, y_n \qquad \limsup_n\, (x_n + y_n)\leq \limsup\, x_n + \limsup\, y_n \] if the additions are not of the type $-\infty + \infty$.

Moreover, the limit of $\{x_n\}$ exists and it is a real number $L$ (respetively $\infty$, $-infty$) if and only if the upper and lower limit coincide and are a real number $L$ (resp. $\infty$, $-\infty$).

The upper and lower limits of a sequence are both finite if and only if the sequence is bounded.

Characterizations

The upper and lower limits can also be defined in several alternative ways. In particular

Theorem 1 Let $S:=\{a\in ]-\infty, \infty] : \{k: x_k >a\} \mbox{is finite}\}$ and $L:= \{a\in [-\infty, \infty[ : \{k: x_k <a\} \mbox{is finite}\}$. Then $\limsup x_n$ is the minimum of $S$ and $\liminf x_n$ is the maximum of $L$.

Theorem 2 Consider the set $A$ of elements $\ell\in [-\infty, \infty]$ for which there is a subsequence of $\{y_n\}$ converging to $\ell$. Then $\limsup x_n$ is the maximum of $A$ and $\liminf x_n$ is the minimum of $A$.

Theorem 3 $U:=\limsup x_n$ is characterized by the two properties:

  • if $U< \infty$ for all $u> U$ there is $N\in \mathbb N$ such that $x_n< u$ for all $n> N$;
  • if $U> -\infty$ for all $u< U$ and $N\in \mathbb N$ there is a $k>N$ with $x_k> u$.

$L:=\liminf x_n$ is characterized by the two properties:

  • if $U> -\infty$ for all $u< U$ there is $N\in\mathbb N$ such that $x_n> u$ for all $n> N$;
  • if $U< \infty$ for all $u> U$ and $N\in\mathbb N$ there is a $k> N$ with $x_k< u$.

Examples

If $x_n = (-1)^n$ then \[ \liminf\, x_n = -1 \qquad \mbox{and} \qquad \limsup_n\, x_n = 1\, . \] If $x_n = (-1)^n n$ then \[ \liminf\, x_n = -\infty \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, . \] If $x_n = n + (-1)^n$, then \[ \liminf\, x_n = 0 \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, . \]

Upper and lower limit of a real function

Definition

If $f$ is a real-valued function defined on a set $E\subset \mathbb R$ (or $\subset \mathbb R^k$), the upper and lower limits of $f$ at $x_0$ are denoted by \[ \limsup_{x\to x_0}\, f(x)\qquad \mbox{and}\qquad \liminf_{x\to x_0}\, f(x)\, . \] and, under the assumptions that $x_0$ is an accumulation point for $E$ (i.e. there is a sequence $\{y_n\}\subset E\setminus x_0\}$ converging to $x_0$) can be defined as

Definition 4 \[ \limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\} \] \[ \liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\} \]

(Some authors include also the point $x_0$ in the definitions above, however this choice is less common).

The definition above can be easilily extended to functions defined on an arbitrary metric space $(X, d)$: it suffices to replace $|x-x_0|< r$ with $d (x, x_0)< r$, namely

Definition 5 \[ \limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\} \] \[ \liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\} \]

As in the case of sequences, some authors use the notation $\overline{\lim}$ and $\underline{\lim}$.

Characterizations

The upper and lower limits \[ U =\limsup_{x\to x_0}\, f(x) \] \[ L :=\liminf_{x\to x_0}\, f(x) \] can also be defined in several alternative ways. A useful one, which reduces to sequences, is the following:

Theorem 6 $U$ is characterized by the properties:

  • There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = U$;
  • For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\limsup\, f (y_k)\leq U$.

$L$ is characterized by the properties:

  • There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = L$;
  • For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\liminf\, f (y_k)\geq L$.

This theorem is valid in an arbitrary metric space.

Properties

From Theorem 6 it can be easily concluded that \[ \liminf_{x\to x_0}\,\, f(x) = - \limsup_{x\to x_0}\, (- f(x)) \] \[ \liminf_{x\to x_0}\,\, (\lambda f (x)) = \lambda\, \liminf_{x\to x_0}\,\, f(x) \qquad \limsup_{x\to x_0}\, (\lambda f(x)) = \lambda\, \limsup_{x\to x_0}\, f(x)\qquad \mbox{when '"`UNIQ-MathJax81-QINU`"'} \] and that \[ \liminf_{x\to x_0}\,\, (f(x) + g(x))\geq \liminf_{x\to x_0}\,\, f(x) + \liminf_{x\to x_0}\,\, g(x) \qquad \limsup_{x\to x_0}\, (f(x) + g(x))\leq \limsup_{x\to x_0}\, f(x) + \limsup_{x\to x_0}\, g(x) \] if the additions are not of the type $-\infty + \infty$.

The function $f$ has a finite limit $L$ (resp. has limit $\infty$, $-\infty$) if the lower and upper limits coincide and are equal to $L$ (resp. $\infty$, $-\infty$).

Moreover, we have the following

Proposition Consider the closed set $E'$ of accumulation points of $E$ and define \[ \underline{f} (x) := \liminf_{y\to x}\,\, f(y)\, \] \[ \overline{f} (x) := \limsup_{y\to x}\, f(y)\, . \] $\underline{f}$ is lower semicontinuous and $\overline{f}$ is upper semicontinuous.

From metric spaces to sequences

Consider the space $X=\mathbb N\cup\{\infty\}$ with the metric: \[ d (n,m) = \left|\frac{1}{m+1} - \frac{1}{n+1}\right|,\qquad d(n,\infty) = \frac{1}{n+1}\, . \] Given a sequence of real numbers $\{x_n\}$ consider the function $f:\mathbb N\to \mathbb R$ given by $f(n) = x_n$ as a function defined on a subset of $X$. Then $\limsup_n\, x_n$ and $\liminf_n\,\, x_n$ in the sense of Definition 1 coincide with \[ \limsup_{n\to\infty}\, f(n)\qquad \liminf_{n\to\infty}\,\, f (n) \] in the metric-space sense of Definition 6.

Upper and lower limit of sets

If $\{A_k\}$ is a sequence of subsets of $X$, the upper and lower limit of the sequence $\{A_k\}$ is defined as \[ \limsup_{k\to\infty}\, A_k = \bigcap_{n\in\mathbb N}\, \bigcup_{k\geq n} A_k\, \] \[ \liminf_{k\to\infty}\, A_k = \bigcup_{n\in\mathbb N}\, \bigcap_{k\geq n} A_k\,. \]

References

[Ap] T.M. Apostol, "Mathematical analysis". Second edition. Addison-Wesley (1974) MR0344384 Zbl 0309.2600
[IlPo] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) MR0687827 Zbl 0138.2730
[Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR0619214 Zbl 0703.26001
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) MR0466435 Zbl 0384.00004
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
How to Cite This Entry:
Upper and lower limits. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Upper_and_lower_limits&oldid=27502
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article