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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the limit inferior and limit superior of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
(see
limit of a function Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
). For a set, they are the
infimum and supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of the set's
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s, 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 In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
) are
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. More generally, these definitions make sense in any
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
, provided the suprema and
infima In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
exist, such as in a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally 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 Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Land ...
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 Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, limit superior and limit inferior are important tools for studying sequences of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. 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 In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
: we add the positive and negative infinities to the real line to give the complete
totally ordered set In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
\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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> is the smallest closed interval with this property. We can formalize this property like this: there exist
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
s 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 In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
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 In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The ter ...
: \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 In a religious context, sin is a transgression against divine law. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered immoral, selfish, s ...
function: x_n = \sin(n). Using the fact that pi is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, 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 In mathematics, the equidistribution theorem is the statement that the sequence :''a'', 2''a'', 3''a'', ... mod 1 is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodic ...
.) * An example from
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
is \liminf_(p_-p_n), where p_n is the n-th
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. The value of this limit inferior is conjectured to be 2 – this is the
twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
– 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 Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of ''f'' at ''0''. This idea of oscillation is sufficient to, for example, characterize
Riemann-integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
functions as continuous except on a set of
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
. 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 In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
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 In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...
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 In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
, is N âˆª .)


Sequences of sets

The
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
℘(''X'') of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''X'' is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
that is ordered by
set inclusion In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
, 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
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
s 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 Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
. 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 In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
(i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
is used to induce the topology on ''X''.


General set convergence

A sequence of sets in a
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric Metric or metrical may refer t ...
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 In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
. 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 In mathematics, the limit of a sequence of sets A_1, A_2, \ldots (subsets of a common set X) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monot ...
. 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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
. 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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
). 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 In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
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 In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
provides a
partial ordering In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
on the collection of all subsets of ''X'' that allows
set intersection In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is writt ...
to generate a greatest lower bound and
set union In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of ze ...
to generate a least upper bound. Thus, the infimum or
meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
of a collection of subsets is the greatest lower bound while the supremum or
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
* The
Borel–Cantelli lemma In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first de ...
is an example application of these constructs. ; Using either the discrete metric or the
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
* 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 In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
and
essential infimum In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''a ...
, 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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s: ::\ = \. :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 In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...
) of a solution to a
dynamic system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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 A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
of several
stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
systems with an undamped second-order
LTI system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined ...
(i.e., zero
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
) 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 A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy ...
. 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 In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of all of the
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s of the set. That is, :\liminf X := \inf \\, Similarly, the limit superior of a set ''X'' is the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
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 In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
''Y'' that is also a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
in order for these definitions to make sense. Moreover, it has to be a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' and a
filter base In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, the ...
''B'' in that space. The set of all
cluster point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s for that filter base is given by :\bigcap \ where \overline_0 is the closure of B_0. This is clearly a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
and is similar to the set of limit points of a set. Assume that ''X'' is also a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
. 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 In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
, is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
and has the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
, :\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
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s. Therefore, these definitions give the limit inferior and
limit superior 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 ...
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 In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. 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 In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''a ...
*
Envelope (waves) In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine ...
*
One-sided limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches ...
*
Dini derivative In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini ...
s *
Set-theoretic limit In mathematics, the limit of a sequence of sets A_1, A_2, \ldots (subsets of a common set X) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monot ...


References

* *


External links

* {{springer, title=Upper and lower limits, id=p/u095830 Limits (mathematics)