# Upper and lower limits

2010 Mathematics Subject Classification: Primary: 54C05 Secondary: 54A05 [MSN][ZBL]

## 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 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 $\liminf_{x\to x_0}\,\, f(x) = - \limsup_{x\to x_0}\, (- f(x))$ $\liminf_{x\to x_0}\,\, (\lambda f (x)) = \lambda\, \liminf_{x\to x_0}\,\, f(x) \qquad \limsup_{x\to x_0}\, (\lambda f(x)) = \lambda\, \limsup_{x\to x_0}\, f(x)\qquad \mbox{when '"UNIQ-MathJax80-QINU"'}$ 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 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 \begin{align*} \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\,. \end{align*} The 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 [Kur].

## 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 [Ku] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) [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://www.encyclopediaofmath.org/index.php?title=Upper_and_lower_limits&oldid=30112
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article