Limsup

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

The limit inferior of a sequence $x_n$ is denoted by
$\liminf_x_n\quad\text\quad \varliminf_x_n.$
The limit superior of a sequence $x_n$ is denoted by
$\limsup_x_n\quad\text\quad \varlimsup_x_n.$

Definition for sequences

The of a sequence (''x''_''n'') is defined by
$\liminf_x_n := \lim_\Big(\inf_x_m\Big)$
or
$\liminf_x_n := \sup_\,\inf_x_m=\sup\.$

Similarly, the of (''x''_''n'') is defined by
$\limsup_x_n := \lim_\Big(\sup_x_m\Big)$
or
$\limsup_x_n := \inf_\,\sup_x_m=\inf\.$

Alternatively, the notations $\varliminf_x_n:=\liminf_x_n$ and $\varlimsup_x_n:=\limsup_x_n$ are sometimes used.

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence $(x_n)$. An element $\xi$ of the extended real numbers $\overline$ is a ''subsequential limit'' of $(x_n)$ if there exists a strictly increasing sequence of natural numbers $(n_k)$ such that $\xi=\lim_ x_$. If $E \subseteq \overline$ is the set of all subsequential limits of $(x_n)$, then

:$\limsup_ x_n = \sup E$

and

:$\liminf_ x_n=\inf E.$

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i. e. the extended real number line) are complete. 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

:$\liminf_x_n\leq\limsup_x_n.$

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 above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior 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. Since the supremum and infimum of an unbounded 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 ˆ’∞,∞ 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 $\varepsilon$, there exists a natural number $N$ such that x_n for all $n>N$. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than $b+\varepsilon$.
* The limit inferior of $x_n$ is the largest real number $b$ such that, for any positive real number $\varepsilon$, there exists a natural number $N$ such that $x_n>b-\varepsilon$ 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 $b-\varepsilon$.

Properties

The relationship of limit inferior and limit superior for sequences of real numbers is as follows:
$\limsup_ \left(-x_n\right) = -\liminf_ x_n$

As mentioned earlier, it is convenient to extend $\R$ to \infty, \infty Then, $\left(x_n\right)$ in \infty, \infty/math> converges if and only if
$\liminf_ x_n = \limsup_ x_n$
in which case $\lim_ x_n$ is equal to their common value. (Note that when working just in $\R,$ convergence to $-\infty$ or $\infty$ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold \begin
\liminf_ x_n &= \infty &&\;\;\text\;\; \lim_ x_n = \infty, \\ .3ex\limsup_ x_n &= - \infty &&\;\;\text\;\; \lim_ x_n = - \infty.
\end

If $I = \liminf_ x_n$ and $S = \limsup_ x_n$, then the interval $$, S/math> need not contain any of the numbers $x_n,$ but every slight enlargement - \epsilon, S + \epsilon for arbitrarily small $\epsilon > 0,$ will contain $x_n$ for all but finitely many indices $n.$ In fact, the interval $$, S/math> is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences $x_$ and $x_$ of $x_n$ (where $k_n$ and $h_n$ are monotonous) for which we have $\liminf_ x_n + \epsilon>x_ \;\;\;\;\;\;\;\;\; x_ > \limsup_ x_n - \epsilon$

On the other hand, there exists a $n_0\in\mathbb$ so that for all $n \geq n_0$ $\liminf_ x_n - \epsilon < x_n < \limsup_ x_n + \epsilon$

To recapitulate:

* If $\Lambda$ is greater than the limit superior, there are at most finitely many $x_n$ greater than $\Lambda;$ if it is less, there are infinitely many.
* If $\lambda$ is less than the limit inferior, there are at most finitely many $x_n$ less than $\lambda;$ if it is greater, there are infinitely many.

In general, $\inf_n x_n \leq \liminf_ x_n \leq \limsup_ x_n \leq \sup_n x_n.$

The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.

* For any two sequences of real numbers $\left\, \left\,$ the limit superior satisfies subadditivity whenever the right side of the inequality is defined (that is, not $\infty - \infty$ or $-\infty + \infty$): $\limsup_ \left(a_n + b_n\right) \leq \limsup_ \left(a_n\right) + \limsup_ \left(b_n\right).$

Analogously, the limit inferior satisfies superadditivity: $\liminf_ \left(a_n + b_n\right) \geq \liminf_\left(a_n\right) + \liminf_\left(b_n\right).$

In the particular case that one of the sequences actually converges, say $a_n \to a,$ then the inequalities above become equalities (with $\limsup_ a_n$ or $\liminf_ a_n$ being replaced by $a$).

* For any two sequences of non-negative real numbers $\left\, \left\,$ the inequalities $\limsup_ (a_nb_n) \leq \left(\limsup_a_n \right) \left(\limsup_ b_n\right)$ and $\liminf_ (a_nb_n) \geq \left(\liminf_a_n \right) \left(\liminf_ b_n\right)$
hold whenever the right-hand side is not of the form $0 \cdot \infty.$

If $\lim_ a_n = A$ exists (including the case $A = +\infty$) and $B = \limsup_ b_n,$ then $\limsup_ \left(a_n b_n\right) = A B$ provided that $A B$ is not of the form $0 \cdot \infty.$

Examples

* As an example, consider the sequence given by the sin function: $x_n = \sin(n).$ Using the fact that pi is irrational, it follows that $\liminf_ x_n = -1$ and $\limsup_ x_n = +1.$ (This is because the sequence $\$ is equidistributed mod 2π, a consequence of the Equidistribution theorem.)

* An example from number theory is $\liminf_(p_-p_n),$ 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 has only been proven to be less than or equal to 246. 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 $\limsup_ f(x) = 1$ and $\liminf_ f(x) = -1$. 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 ''0''. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. 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 topological spaces to complete lattices

Functions from 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 a metric space $X$, a subspace $E$ contained in $X$, and a function $f:E \to \mathbb$. Define, for any limit point $a$ of $E$,

:$\limsup_ f(x) = \lim_ ( \sup \ )$
and

:$\liminf_ f(x) = \lim_ ( \inf \ )$

where $B(a,\varepsilon)$ denotes the metric ball of radius $\varepsilon$ about $a$.

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

:$\limsup_ f(x) = \inf_ (\sup \)$
and similarly
:$\liminf_ f(x) = \sup_(\inf \).$

Functions from topological spaces

This finally motivates the definitions for general topological spaces. Take ''X'', ''E'' and ''a'' as before, but now let ''X'' be a topological space. In this case, we replace metric balls with neighborhoods:

:$\limsup_ f(x) = \inf \$
:$\liminf_ f(x) = \sup \$

(there is a way to write the formula using "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 ˆ’∞,∞ the extended real number line, is N âˆª .)

Sequences of sets

The power set ℘(''X'') of a set ''X'' is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset ''Y'' of ''X'' is bounded above by ''X'' and below by the empty set ∅ because ∅ ⊆ ''Y'' ⊆ ''X''. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(''X'') (i.e., sequences of subsets of ''X'').

There are two common ways to define the limit of sequences of sets. In both cases:
* The sequence ''accumulates'' around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation ''sets'' that are somehow nearby to infinitely many elements of the sequence.
* The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it ''contains'' each of them. Hence, it is the supremum of the limit points.
* The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is ''contained in'' each of them. Hence, it is the infimum of the limit points.
* Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf ''X''_''n'' ⊆ lim sup ''X''_''n''). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence.
The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on ''X''.

General set convergence

A sequence of sets in a metrizable space $X$ approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if $\left\$ is a sequence of subsets of $X,$ then:
* $\limsup X_n,$ which is also called the outer limit, consists of those elements which are limits of points in $X_n$ taken from (countably) infinitely many $n.$ That is, $x \in \limsup X_n$ if and only if there exists a sequence of points $\left\$ and a $\left\$ of $\left\$ such that $x_k \in X_$ and $\lim_ x_k = x.$
* $\liminf X_n,$ which is also called the inner limit, consists of those elements which are limits of points in $X_n$ for all but finitely many $n$ (that is, cofinitely many $n$). That is, $x \in \liminf X_n$ if and only if there exists a of points $\left\$ such that $x_k \in X_k$ and $\lim_ x_k =x.$
The limit $\lim X_n$ exists if and only if $\,\liminf X_n \text \limsup X_n\,$ agree, in which case $\,\lim X_n = \limsup X_n = \liminf X_n.$ The outer and inner limits should not be confused with the set-theoretic limits superior and inferior, as the latter sets are not sensitive to the topological structure of the space.

Special case: discrete metric

This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at set-theoretic limit.

By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence ''and'' does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set ''X'' is induced from the discrete metric.

Specifically, for points ''x'' ∈ ''X'' and ''y'' ∈ ''X'', the discrete metric is defined by
:$d(x,y) := \begin 0 &\text x = y,\\ 1 &\text x \neq y, \end$
under which a sequence of points converges to point ''x'' ∈ ''X'' if and only if ''x''_''k'' = ''x'' for all except finitely many ''k''. Therefore, ''if the limit set exists'' it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.

If is a sequence of subsets of ''X'', then the following always exist:
* lim sup ''X''_''n'' consists of elements of ''X'' which belong to ''X''_''n'' for infinitely many ''n'' (see countably infinite). That is, ''x'' ∈ lim sup ''X''_''n'' if and only if there exists a subsequence of such that ''x'' ∈ ''X''_''n''''k'' for all ''k''.
* lim inf ''X''_''n'' consists of elements of ''X'' which belong to ''X''_''n'' for all except finitely many ''n'' (i.e., for cofinitely many ''n''). That is, ''x'' ∈ lim inf ''X''_''n'' if and only if there exists some ''m''>0 such that ''x'' ∈ ''X''_''n'' for all ''n''>''m''.
Observe that ''x'' ∈ lim sup ''X''_''n'' if and only if ''x'' ∉ lim inf ''X''_''n''^c.
* The lim ''X''_''n'' exists if and only if lim inf ''X''_''n'' and lim sup ''X''_''n'' agree, in which case lim ''X''_''n'' = lim sup ''X''_''n'' = lim inf ''X''_''n''.
In this sense, the sequence has a limit so long as every point in ''X'' either appears in all except finitely many ''X''_''n'' or appears in all except finitely many ''X''_''n''^c.

Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of ''X'' that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf ''X''_''n'', is the largest meeting of tails of the sequence, and the outer limit, lim sup ''X''_''n'', is the smallest joining of tails of the sequence. The following makes this precise.
*Let ''I''_''n'' be the meet of the ''n''^th tail of the sequence. That is,
::$\beginI_n &= \inf \\\ &= \bigcap_^ X_m = X_n \cap X_ \cap X_ \cap \cdots. \end$
:The sequence is non-decreasing (''I''_''n'' ⊆ ''I''_''n''+1) because each ''I''_''n''+1 is the intersection of fewer sets than ''I''_''n''. The least upper bound on this sequence of meets of tails is
::$\begin \liminf_X_n &= \sup\\\ &= \left(X_m\right). \end$
:So the limit infimum contains all subsets which are lower bounds for all except finitely many sets of the sequence.

*Similarly, let ''J''_''n'' be the join of the ''n''^th tail of the sequence. That is,
::$\beginJ_n &= \sup \\\ &= \bigcup_^ X_m = X_n \cup X_ \cup X_ \cup \cdots. \end$
:The sequence is non-increasing (''J''_''n'' ⊇ ''J''_''n''+1) because each ''J''_''n''+1 is the union of fewer sets than ''J''_''n''. The greatest lower bound on this sequence of joins of tails is
::$\begin \limsup_X_n &= \inf\\\ &= \left(X_m\right). \end$
:So the limit supremum is contained in all subsets which are upper bounds for all except finitely many sets of the sequence.

Examples

The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set ''X''.

; Using the discrete metric

* The Borel–Cantelli lemma is an example application of these constructs.

; Using either the discrete metric or the Euclidean metric

* Consider the set ''X'' = and the sequence of subsets:
::$\ = \.$
:The "odd" and "even" elements of this sequence form two subsequences, and , which have limit points 0 and 1, respectively, and so the outer or superior limit is the set of these two points. However, there are no limit points that can be taken from the sequence as a whole, and so the interior or inferior limit is the empty set . That is,
:* lim sup ''X''_''n'' =
:* lim inf ''X''_''n'' =
:However, for = and = :
:* lim sup ''Y''_''n'' = lim inf ''Y''_''n'' = lim ''Y''_''n'' =
:* lim sup ''Z''_''n'' = lim inf ''Z''_''n'' = lim ''Z''_''n'' =

* Consider the set ''X'' = and the sequence of subsets:
::$\ = \.$
:As in the previous two examples,
:* lim sup ''X''_''n'' =
:* lim inf ''X''_''n'' =
:That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence (e.g., at positions 100, 150, 275, and 55000). So long as the ''tails'' of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of ''essential'' inner and outer limits, which use the essential supremum and essential infimum, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.

; Using the Euclidean metric

* Consider the sequence of subsets of rational numbers:
::$\ = \.$
:The "odd" and "even" elements of this sequence form two subsequences, and , which have limit points 1 and 0, respectively, and so the outer or superior limit is the set of these two points. However, there are no limit points that can be taken from the sequence as a whole, and so the interior or inferior limit is the empty set . So, as in the previous example,
:* lim sup ''X''_''n'' =
:* lim inf ''X''_''n'' =
:However, for = and = :
:* lim sup ''Y''_''n'' = lim inf ''Y''_''n'' = lim ''Y''_''n'' =
:* lim sup ''Z''_''n'' = lim inf ''Z''_''n'' = lim ''Z''_''n'' =
:In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.

* The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system. Because trajectories become closer and closer to this limit set, the tails of these trajectories ''converge'' to the limit set.
:* For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the state space. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.

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'' ⊆ ''Y'' is the infimum of all of the limit points of the set. That is,
:$\liminf X := \inf \\,$
Similarly, the limit superior of a set ''X'' is the supremum of all of the limit points of the set. That is,
:$\limsup X := \sup \\,$
Note that the set ''X'' needs to be defined as a subset of a partially ordered set ''Y'' 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 the limit inferior and the limit superior of a set do not have to be elements 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
:$\bigcap \$
where $\overline_0$ 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 superior of the filter base ''B'' is defined as
:$\limsup B := \sup \bigcap \$
when that supremum exists. When ''X'' has a total order, is a complete lattice and has the order topology,
:$\limsup B = \inf\.$
Similarly, the limit inferior of the filter base ''B'' is defined as
:$\liminf B := \inf \bigcap \$
when that infimum exists; if ''X'' is totally ordered, is a complete lattice, and has the order topology, then
:$\liminf B = \sup\.$

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 $(x_\alpha)_$, where $(A,)$ is a directed set and $x_\alpha \in X$ for all $\alpha \in A$. The filter base ("of tails") generated by this net is $B$ defined by
:$B := \.\,$
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 $n \in \mathbb$ with $\mathbb$ being the set of natural numbers. The filter base ("of tails") generated by this sequence is $C$ defined by
:$C := \.\,$
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 infimum and essential supremum
* Envelope (waves)
* One-sided limit
* Dini derivatives
* Set-theoretic limit

References

*
*

External links

* {{springer, title=Upper and lower limits, id=p/u095830

Limits (mathematics)