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In mathematics, the 'limit inferior' and 'limit superior' (also called 'infimum limit' and 'supremum limit', or 'liminf' and 'limsup') of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. The limit inferior and limit superior of a function can be thought of in a similar fashion. (See limit of a function.) The limit inferior and limit superior of a set is the infimum and supremum of the set's limit points respectively.

Contents

Definition for sequences

The case of sequences of real numbers

Interpretation

Properties

Real-valued functions

Functions from metric spaces to metric spaces

Sequences of sets

Generalized definitions

Definition for a set

Definition for filter bases

Specialization for sequences and nets

See also

References

Definition for sequences

The limit inferior of a sequence (''x''_''n'') is defined as
:$liminf\_\{n$
ightarrowinfty}x_n=sup_{ngeq 0},inf_{mgeq n}x_m=sup{,inf{,x_m:mgeq n,}:ngeq 0,}
or
:$liminf\_\{n$
ightarrowinfty}x_n=lim_{n
ightarrowinfty}Big(inf_{mgeq n}x_mBig).
Similarly, the limit superior of (''x''_''n'') is defined as
:$limsup\_\{n$
ightarrowinfty}x_n=inf_{ngeq 0},sup_{mgeq n}x_m=inf{,sup{,x_m:mgeq n,}:ngeq 0,}
or
:$limsup\_\{n$
ightarrowinfty}x_n=lim_{n
ightarrowinfty}Big(sup_{mgeq n}x_mBig).
If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.
Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does ''not'' exist. Whenever lim inf ''x''_''n'' and lim sup ''x''_''n'' both exist, we have
:$liminf\_\{n$
ightarrowinfty}x_nleqlimsup_{n
ightarrowinfty}x_n.
Limits inferior/superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e^-''n'' may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded by the limit superior (inferior) plus (minus) an arbitrarily small positive constant.
The limit superior and limit inferior of a sequence are a special case of those of a function, see below.

The case of sequences of real numbers

In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. In order to deal with the difficulties arising from the fact that the supremum and infimum of a set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [-∞,∞], which is a complete lattice.

Interpretation

Consider a sequence $(x\_n)$ consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).

★ The limit superior of $x\_n$ is the smallest real number $b$ such that, for any positive real number , there exists a natural number $N$ such that

★ The limit inferior of $x\_n$ is the largest real number $b$ that, for any positive real number , there exists a natural number $N$ such that for all $n>N$. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than this lower bound.

Properties

The relationship of limit inferior and limit superior for sequences of real numbers is as follows
:$-liminf\_\{n\; oinfty\}\; x\_n\; =\; limsup\_\{n\; oinfty\}\; (-x\_n).$
As mentioned earlier, it is convenient to extend 'R' to [-∞,∞]. Then, (''x''_''n'') in [-∞,∞] converges if and only if
:$liminf\_\{n\; oinfty\}\; x\_n\; =\; limsup\_\{n\; oinfty\}\; x\_n,$
in which case
:$lim\_\{n\; oinfty\}\; x\_n$
is equal to their common value. (Note that when working just in 'R', convergence to -∞ or ∞ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the condition
:$liminf\_\{n\; oinfty\}\; x\_n\; =\; infty$
implies that
:$lim\_\{n\; oinfty\}\; x\_n\; =\; infty,$
and the condition
:$limsup\_\{n\; oinfty\}\; x\_n\; =\; -\; infty$
implies that
:$lim\_\{n\; oinfty\}\; x\_n\; =\; -\; infty$.
As an example, consider the sequence given by ''x''_''n'' = sin(''n''). Using the fact that pi is irrational, one can show that
:$liminf\_\{n\; oinfty\}\; x\_n\; =\; -1$
and
:$limsup\_\{n\; oinfty\}\; x\_n\; =\; +1.$
(This is because the sequence {1,2,3,...} is equidistributed mod 2π, a consequence of the Equidistribution theorem.)
If
:$I\; =\; liminf\_\{n\; oinfty\}\; x\_n$
and
:$S\; =\; limsup\_\{n\; oinfty\}\; x\_n,$
then the interval [''I'', ''S''] need not contain any of the numbers ''x''_''n'', but every slight enlargement [''I'' − ε, ''S'' + ε] (for arbitrarily small ε > 0) will contain ''x''_''n'' for all but finitely many indices ''n''. In fact, the interval [''I'', ''S''] is the smallest closed interval with this property.
An example from number theory is
:$liminf\_\{n\; oinfty\}(p\_\{n+1\}-p\_n),$
where ''p''_''n'' is the ''n''-th prime number.
The value of this limit inferior is conjectured to be 2 - this is the twin prime conjecture - but as yet has not even been proved finite. The corresponding limit superior is $+infty$, because there are arbitrary gaps between consecutive primes.

Real-valued functions

Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and -∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given ''f''(''x'') = sin(1/''x''), we have lim sup_{''x''→''0''} ''f''(''x'') = 1 and lim inf_{''x''→''0''} ''f''(''x'') = -1. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of ''f'' at ''a''. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero [1]. Note that points of nonzero oscillation (i.e., points at which ''f'' is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

Functions from metric spaces to metric spaces

There is a notion of lim sup and lim inf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Take metric spaces ''X'' and ''Y'', a subspace ''E'' contained in ''X'', and function ''f'': ''E'' → ''Y''. Define for any limit point ''a'' of ''E'':
:
and
:
where ''B''(''a'';ε) denotes the metric ball of radius ε about ''a''.
Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have
:
and similarly
:
This finally motivates the definitions for general topological spaces. Take ''X'', ''Y'', ''E'' and ''a'' as before, but now let ''X'' and ''Y'' both be topological spaces. In this case, we replace metric balls with neighborhoods:
:$limsup\_\{x\; o\; a\}\; f(x)\; =\; inf\; \{\; sup\; \{\; f(x)\; :\; x\; in\; E\; cap\; U\; -\; \{a\}\; \}\; :\; U\; mathrm\{open\},\; a\; in\; U,\; E\; cap\; U\; -\; \{a\}$
eq emptyset }
:$liminf\_\{x\; o\; a\}\; f(x)\; =\; sup\; \{\; inf\; \{\; f(x)\; :\; x\; in\; E\; cap\; U\; -\; \{a\}\; \}\; :\; U\; mathrm\{open\},\; a\; in\; U,\; E\; cap\; U\; -\; \{a\}$
eq emptyset }
(there is a way to write the formula using a ''lim'' using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of 'N' in [-∞, ∞] is 'N' ∪ {∞}.)

Sequences of sets

The power set ''P''(''X'') of a set ''X'' is a complete lattice, and it is sometimes useful to consider limits superior and inferior of sequences in ''P''(''X''), that is, sequences of subsets of ''X''.
If ''X''_''n'' is such a sequence, then an element ''a'' of ''X'' belongs to lim inf ''X''_''n'' if and only if there exists a natural number ''n''₀ such that ''a'' is in ''X''_''n'' for all ''n'' > ''n''₀. The element ''a'' belongs to lim sup ''X''_''n'' if and only if for every natural number ''n''₀ there exists an index ''n'' > ''n''₀ such that ''a'' is in ''X''_''n''.
In other words, lim sup ''X''_''n'' consists of those elements which are in ''X''_''n'' for infinitely many ''n'', while lim inf ''X''_''n'' consists of those elements which are in ''X''_''n'' for all but finitely many ''n''.
As an example, consider the sequence
:$\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},dots,$
whose superior limit is {0,1} but whose inferior limit is empty.
Using the standard parlance of set theory, the infimum of a sequence of sets is the countable intersection of the sets, the largest set included in all of the sets:
:$infleft\{,x\_n\; :\; n=1,2,3,dots,$
ight}={igcap_{n=1}^infty}x_n.
The sequence of ''I''_''n'', ''n'' = 1, 2, 3, ..., where ''I''_''n'' is the infimum of set ''n'', is non-decreasing, because ''I''_''n'' ⊂ ''I''_''n''+1. Therefore, the countable union of infimum from 1 to n is equal to the nth infimum. Taking this sequence of sets to the limit:
:$liminf\_\{n$
ightarrowinfty}x_n={igcup_{n=1}^infty}left({igcap_{m=n}^infty}x_m
ight).
The limsup can be defined in a dual fashion. The supremum of a sequence of sets is the smallest set containing all the sets, i.e., the countable union of the sets.
:$supleft\{,x\_n\; :\; n=1,2,3,dots,$
ight}={igcup_{n=1}^infty}x_n.
The limsup is the countable intersection of this non-increasing (each supremum is a subset of the previous supremum) sequence of sets.
:$limsup\_\{n$
ightarrowinfty}x_n={igcap_{n=1}^infty}left({igcup_{m=n}^infty}x_m
ight).
See Borel-Cantelli lemma for an example.

Generalized definitions

The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

Definition for a set

The limit inferior of a set ''X'' is the infimum of all of the limit points of the set. That is,
:$liminf\; X\; =\; inf\; \{\; x\; in\; X\; :\; x\; ext\{\; is\; a\; limit\; point\; of\; \}\; X\; \},$
Similarly, the limit superior of a set ''X'' is the supremum of all of the limit points of the set. That is,
:$limsup\; X\; =\; sup\; \{\; x\; in\; X\; :\; x\; ext\{\; is\; a\; limit\; point\; of\; \}\; X\; \},$
Note that the set ''X'' needs to be defined as a subset of a partially ordered set that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice, so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior.
Also note that neither the limit inferior nor the limit superior of a set must be an element of the set.

Definition for filter bases

Take a topological space ''X'' and a filter base ''B'' in that space. The set of all cluster points for that filter base is given by
:
where $overline\{B\}\_0$ is the closure of $B\_0$. This is clearly a closed set and is similar to the set of limit points of a set. Assume that ''X'' is also a partially ordered set. The limit inferior of the filter base ''B'' is defined as
:
and can be thought of as the smallest cluster point of the filter base. Similarly, the limit superior of the filter base ''B'' is defined as
:
and can be thought of as the largest cluster point of the filter base. If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

Specialization for sequences and nets

Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space $X$ and the net , where $(A,\{leq\})$ is a directed set and for all . The filter base ("of tails") generated by this net is $B$ defined by
:
Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of $B$ respectively. Similarly, for topological space $X$, take the sequence $(x\_n)$ where $x\_n\; in\; X$ for any $n\; in\; mathbb\{N\}$ with $mathbb\{N\}$ being the set of natural numbers. The filter base ("of tails") generated by this sequence is $C$ defined by
:$C\; riangleq\; \{\; \{\; x\_n\; :\; n\_0\; leq\; n\; \}\; :\; n\_0\; in\; mathbb\{N\}\; \}.,$
Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of $C$ respectively.

See also

★ Essential supremum and essential infimum

References

★ Analysis, , H., Amann, Basel; Boston: Birkhäuser, ,

★ Classical complex analysis, , Mario O, González, New York: M. Dekker, ,