View source for Upper and lower limits

{{MSC|54C05|54A05}}

{{TEX|done}}

==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
\[
\liminf_n\,\, x_n = -\limsup_n\, (-x_n)\, , 
\]
\[
\liminf_n\,\, (\lambda x_n) = \lambda\, \liminf_n\,\, x_n\qquad \limsup_n\, (\lambda x_n) = \lambda\, \limsup_n\, x_n\qquad \mbox{when $\lambda > 0$}
\]
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 $-\infty + \infty$.

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|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|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
\[
\liminf_{x\to x_0}\,\, f(x) = - \limsup_{x\to x_0}\, (- f(x))
\]
\[
\liminf_{x\to x_0}\,\, (\lambda f (x)) = \lambda\,  \liminf_{x\to x_0}\,\, f(x) \qquad \limsup_{x\to x_0}\, (\lambda f(x)) = \lambda\, \limsup_{x\to x_0}\, f(x)\qquad \mbox{when $\lambda > 0$}
\]
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 $-\infty + \infty$.

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 [[Semi-continuous function|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 in set theory==
If $\{A_k\}$ is a sequence of subsets of $X$, the upper and lower limit of the sequence $\{A_k\}$ are defined as
\[
\limsup_{k\to\infty}\, A_k = \bigcap_{n\in\mathbb N}\, \bigcup_{k\geq n} A_k\,,
\]
\[
\liminf_{k\to\infty}\, A_k = \bigcup_{n\in\mathbb N}\, \bigcap_{k\geq n} A_k\,.
\]
The [[Characteristic function of a set|indicator function]] of the upper limit is the (pointwise) upper limit of indicator functions $\bsone_{A_n}(\cdot)$. The same holds for the lower limit.

==Lower limit of sets in topology==
If $\{A_i\}$ is a sequence of subsets of a topological space $X$, the terminology ''lower limit'' is also used for the set of those points $p\in X$ with the property that for every neighborhood $U$ of $p$ there is an $N$ with $A_i\cap U\neq \emptyset$ $\forall i\geq N$. See for instance {{Cite|Kur}}.

==References==
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|valign="top"|{{Ref|Ku}}||valign="top"|  L.D. Kudryavtsev,   "Mathematical analysis" , '''1''' , Moscow  (1973)   (In Russian) {{MR|0619214}} {{ZBL|0703.26001}}
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|}