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As the positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit of a sequence is the value that the terms 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 ...
"tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of
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 ...
ultimately rests. Limits can be defined in any
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
or
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 po ...
, but are usually first encountered in the
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 ...
s.


History

The Greek philosopher
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
is famous for formulating paradoxes that involve limiting processes.
Leucippus Leucippus (; el, Λεύκιππος, ''Leúkippos''; fl. 5th century BCE) is a pre-Socratic Greek philosopher who has been credited as the first philosopher to develop a theory of atomism. Leucippus' reputation, even in antiquity, was obscured ...
,
Democritus Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe. No ...
,
Antiphon An antiphon ( Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christian ritual, sung as a refrain. The texts of antiphons are the Psalms. Their form was favored by St Ambrose and they feature prominentl ...
, Eudoxus, and Archimedes developed the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
.
Grégoire de Saint-Vincent Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of th ...
gave the first definition of limit (terminus) of a
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment." Newton dealt with series in his works on ''Analysis with infinite series'' (written in 1669, circulated in manuscript, published in 1711), ''Method of fluxions and infinite series'' (written in 1671, published in English translation in 1736, Latin original published much later) and ''Tractatus de Quadratura Curvarum'' (written in 1693, published in 1704 as an Appendix to his ''Optiks''). In the latter work, Newton considers the binomial expansion of (''x'' + ''o'')''n'', which he then linearizes by ''taking the limit'' as ''o'' tends to 0. In the 18th century,
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s such as Euler succeeded in summing some ''divergent'' series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus.
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in his etude of hypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit. The modern definition of a limit (for any ε there exists an index ''N'' so that ...) was given by
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his lib ...
(''Der binomische Lehrsatz'', Prague 1816, which was little noticed at the time), and by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
in the 1870s.


Real numbers

In the real numbers, a number L is the limit of the
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 ...
(x_n), if the numbers in the sequence become closer and closer to L, and not to any other number.


Examples

*If x_n = c for constant ''c'', then x_n \to c.''Proof'': Choose N = 1. For every n \geq N, , x_n - c, = 0 < \varepsilon *If x_n = \frac, then x_n \to 0.''Proof'': choose N = \left\lfloor\frac\right\rfloor + 1 (the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
). For every n \geq N, , x_n - 0, \le x_N = \frac < \varepsilon.
*If x_n = \frac when n is even, and x_n = \frac when n is odd, then x_n \to 0. (The fact that x_ > x_n whenever n is odd is irrelevant.) *Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence 0.3, 0.33, 0.333, 0.3333, \dots converges to 1/3. Note that the
decimal representation A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, i ...
0.3333\dots is the ''limit'' of the previous sequence, defined by 0.3333... : = \lim_ \sum_^n \frac *Finding the limit of a sequence is not always obvious. Two examples are \lim_ \left(1 + \tfrac\right)^n (the limit of which is the number ''e'') and the Arithmetic–geometric mean. The
squeeze theorem In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions. The squeeze theorem is used in calculus and mathematical anal ...
is often useful in the establishment of such limits.


Definition

We call x the limit of the
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 ...
(x_n), which is written :x_n \to x, or :\lim_ x_n = x, if the following condition holds: :For each
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 ...
\varepsilon > 0, 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 such that, for every natural number n \geq N, we have , x_n - x, < \varepsilon. In other words, for every measure of closeness \varepsilon, the sequence's terms are eventually that close to the limit. The sequence (x_n) is said to converge to or tend to the limit x. Symbolically, this is: :\forall \varepsilon > 0 \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies , x_n - x, < \varepsilon \right)\right)\right). If a sequence (x_n) converges to some limit x, then it is convergent and x is the only limit; otherwise (x_n) is divergent. A sequence that has zero as its limit is sometimes called a null sequence.


Illustration

File:Folgenglieder im KOSY.svg, Example of a sequence which converges to the limit a. File:Epsilonschlauch.svg, Regardless which \varepsilon > 0 we have, there is an index N_0, so that the sequence lies afterwards completely in the epsilon tube (a-\varepsilon,a+\varepsilon). File:Epsilonschlauch klein.svg, There is also for a smaller \varepsilon_1 > 0 an index N_1, so that the sequence is afterwards inside the epsilon tube (a-\varepsilon_1,a+\varepsilon_1). File:Epsilonschlauch2.svg, For each \varepsilon > 0 there are only finitely many sequence members outside the epsilon tube.


Properties

Some other important properties of limits of real sequences include the following: *When it exists, the limit of a sequence is unique. *Limits of sequences behave well with respect to the usual
arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
. If \lim_ a_n and \lim_ b_n exists, then ::\lim_ (a_n \pm b_n) = \lim_ a_n \pm \lim_ b_n ::\lim_ c a_n = c \cdot \lim_ a_n ::\lim_ (a_n \cdot b_n) = \left(\lim_ a_n \right)\cdot \left( \lim_ b_n \right) ::\lim_ \left(\frac\right) = \frac provided \lim_ b_n \ne 0 ::\lim_ a_n^p = \left( \lim_ a_n \right)^p *For any continuous function ''f'', if \lim_x_n exists, then \lim_ f \left(x_n \right) exists too. In fact, any real-valued
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 ...
''f'' is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity). *If a_n \leq b_n for all n greater than some N, then \lim_ a_n \leq \lim_ b_n. *(
Squeeze theorem In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions. The squeeze theorem is used in calculus and mathematical anal ...
) If a_n \leq c_n \leq b_n for all n greater than some N, and \lim_ a_n = \lim_ b_n = L, then \lim_ c_n = L. *(
Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Infor ...
) If a_n is bounded and
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
for all n greater than some N, then it is convergent. *A sequence is convergent if and only if every subsequence is convergent. *If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point. These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example, once it is proven that 1/n \to 0, it becomes easy to show—using the properties above—that \frac \to \frac (assuming that b \ne 0).


Infinite limits

A sequence (x_n) is said to tend to infinity, written :x_n \to \infty, or :\lim_x_n = \infty, if the following holds: :For every real number K, there is a natural number N such that for every natural number n \geq N, we have x_n > K; that is, the sequence terms are eventually larger than any fixed K. Symbolically, this is: :\forall K \in \mathbb \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n > K \right)\right)\right). Similarly, we say a sequence tends to minus infinity, written :x_n \to -\infty, or :\lim_x_n = -\infty, if the following holds: :For every real number K, there is a natural number N such that for every natural number n \geq N, we have x_n < K; that is, the sequence terms are eventually smaller than any fixed K. Symbolically, this is: :\forall K \in \mathbb \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies x_n < K \right)\right)\right). If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence x_n=(-1)^n provides one such example.


Metric spaces


Definition

A point x of the
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 set ...
(X, d) is the limit of the
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 ...
(x_n) if: :For each
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 ...
\varepsilon > 0, there is 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 such that, for every natural number n \geq N, we have d(x_n, x) < \varepsilon . Symbolically, this is: :\forall \varepsilon > 0 \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies d(x_n, x) < \varepsilon \right)\right)\right). This coincides with the definition given for real numbers when X = \R and d(x, y) = , x-y, .


Properties

*When it exists, the limit of a sequence is unique, as distinct points are separated by some positive distance, so for \varepsilon less than half this distance, sequence terms cannot be within a distance \varepsilon of both points. *For any continuous function ''f'', if \lim_ x_n exists, then \lim_ f(x_n) = f\left(\lim_x_n \right). In fact, 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 ...
''f'' is continuous if and only if it preserves the limits of sequences.


Cauchy sequences

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in
metric spaces 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 ...
, and, in particular, in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
. One particularly important result in real analysis is the ''Cauchy criterion for convergence of sequences'': a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
s.


Topological spaces


Definition

A point x \in X of the topological space (X, \tau) is a or of the
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 ...
\left(x_n\right)_ if: :For every
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
U of x, there exists some N \in \N such that for every n \geq N, we have x_n \in U. This coincides with the definition given for metric spaces, if (X, d) is a metric space and \tau is the topology generated by d. A limit of a sequence of points \left(x_n\right)_ in a topological space T is a special case of a limit of a function: the domain is \N in the space \N \cup \lbrace + \infty \rbrace, with the
induced topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
of 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 on ...
, the range is T, and the function argument n tends to +\infty, which in this space is a
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 ...
of \N.


Properties

In a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points x and y are
topologically indistinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhood (topology), neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborh ...
, then any sequence that converges to x must converge to y and vice versa.


Hyperreal numbers

The definition of the limit using the
hyperreal numbers In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence (x_n) tends to ''L'' if for every infinite hypernatural ''H'', the term x_H is infinitely close to ''L'' (i.e., the difference x_H - L is infinitesimal). Equivalently, ''L'' is the
standard part In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
of x_H: : L = (x_H). Thus, the limit can be defined by the formula :\lim_ x_n= (x_H). where the limit exists if and only if the righthand side is independent of the choice of an infinite ''H''.


Sequence of more than one index

Sometimes one may also consider a sequence with more than one index, for example, a double sequence (x_). This sequence has a limit L if it becomes closer and closer to L when both ''n'' and ''m'' becomes very large.


Example

*If x_ = c for constant ''c'', then x_ \to c. *If x_ = \frac, then x_ \to 0. *If x_ = \frac, then the limit does not exist. Depending on the relative "growing speed" of ''n'' and ''m'', this sequence can get closer to any value between 0 and 1.


Definition

We call x the double limit of the
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 ...
(x_), written :x_ \to x, or :\lim_ x_ = x, if the following condition holds: :For each
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 ...
\varepsilon > 0, 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 such that, for every pair of natural numbers n, m \geq N, we have , x_ - x, < \varepsilon. In other words, for every measure of closeness \varepsilon, the sequence's terms are eventually that close to the limit. The sequence (x_) is said to converge to or tend to the limit x. Symbolically, this is: :\forall \varepsilon > 0 \left(\exists N \in \N \left(\forall n, m \in \N \left(n, m \geq N \implies , x_ - x, < \varepsilon \right)\right)\right) . Note that the double limit is different from taking limit in ''n'' first, and then in ''m''. The latter is known as
iterated limit In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form : \lim_ \lim_ a_ = \lim_ \left( \lim_ a_ \right), : \lim_ \lim_ f(x, y) = \lim_ \left( \lim_ f(x, y) \right), or other similar forms. An ...
. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.


Infinite limits

A sequence (x_) is said to tend to infinity, written :x_ \to \infty, or :\lim_x_ = \infty, if the following holds: :For every real number K, there is a natural number N such that for every pair of natural numbers n,m \geq N, we have x_ > K; that is, the sequence terms are eventually larger than any fixed K. Symbolically, this is: :\forall K \in \mathbb \left(\exists N \in \N \left(\forall n, m \in \N \left(n, m \geq N \implies x_ > K \right)\right)\right). Similarly, a sequence (x_) tends to minus infinity, written :x_ \to -\infty, or :\lim_x_ = -\infty, if the following holds: :For every real number K, there is a natural number N such that for every pair of natural numbers n,m \geq N, we have x_ < K; that is, the sequence terms are eventually smaller than any fixed K. Symbolically, this is: :\forall K \in \mathbb \left(\exists N \in \N \left(\forall n, m \in \N \left(n, m \geq N \implies x_ < K \right)\right)\right). If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence x_=(-1)^ provides one such example.


Pointwise limits and uniform limits

For a double sequence (x_), we may take limit in one of the indices, say, n \to \infty, to obtain a single sequence (y_m). In fact, there are two possible meanings when taking this limit. The first one is called pointwise limit, denoted :x_ \to y_m\quad \text, or :\lim_ x_ = y_m\quad \text, which means: :For each
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 ...
\varepsilon > 0 and each fixed
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 ...
m, there exists a natural number N(\varepsilon, m) > 0 such that, for every natural number n \geq N, we have , x_ - y_m, < \varepsilon. Symbolically, this is: :\forall \varepsilon > 0 \left( \forall m \in \mathbb \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies , x_ - y_m, < \varepsilon \right)\right)\right)\right). When such a limit exists, we say the sequence (x_) converges pointwise to (y_m). The second one is called uniform limit, denoted :x_ \to y_m \quad \text, :\lim_ x_ = y_m \quad \text, :x_ \rightrightarrows y_m , or :\underset \; x_ = y_m , which means: :For each
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 ...
\varepsilon > 0, there exists a natural number N(\varepsilon) > 0 such that, for every
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 ...
m and for every natural number n \geq N, we have , x_ - y_m, < \varepsilon. Symbolically, this is: :\forall \varepsilon > 0 \left(\exists N \in \N \left( \forall m \in \mathbb \left(\forall n \in \N \left(n \geq N \implies , x_ - y_m, < \varepsilon \right)\right)\right)\right). In this definition, the choice of N is independent of m. In other words, the choice of N is ''uniformly applicable'' to all natural numbers m. Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit: :If x_ \to y_m uniformly, then x_ \to y_m pointwise. When such a limit exists, we say the sequence (x_) converges uniformly to (y_m).


Iterated limit

For a double sequence (x_), we may take limit in one of the indices, say, n \to \infty, to obtain a single sequence (y_m), and then take limit in the other index, namely m \to \infty, to get a number y. Symbolically, :\lim_ \lim_ x_ = \lim_ y_m = y. This limit is known as
iterated limit In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form : \lim_ \lim_ a_ = \lim_ \left( \lim_ a_ \right), : \lim_ \lim_ f(x, y) = \lim_ \left( \lim_ f(x, y) \right), or other similar forms. An ...
of the double sequence. Note that the order of taking limits may affect the result, i.e., :\lim_ \lim_ x_ \ne \lim_ \lim_ x_ in general. A sufficient condition of equality is given by the
Moore-Osgood theorem In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form : \lim_ \lim_ a_ = \lim_ \left( \lim_ a_ \right), : \lim_ \lim_ f(x, y) = \lim_ \left( \lim_ f(x, y) \right), or other similar forms. An ...
, which requires the limit \lim_x_ = y_m to be uniform in ''m''.


See also

*
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 ...
*
Subsequential limit In mathematics, a subsequential limit of a sequence is the Limit of a sequence, limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In First-countable space, first-countable spaces, the two concepts coincid ...
*
Limit superior and limit inferior 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 ...
* Limit of a function * Limit of a sequence of functions * Limit of a sequence of sets *
Limit of a net In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codoma ...
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Pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
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Uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
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Modes of convergence In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, se ...


Notes


Proofs


References

* * * Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow. *
Frank Morley Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebr ...
and James Harknessbr>A treatise on the theory of functions
(New York: Macmillan, 1893)


External links

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{{Calculus topics Limits (mathematics) Sequences and series