TheInfoList

OR:

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 language o ...
$n$ becomes larger and larger, the value $n\cdot \sin\left\left(\tfrac1\right\right)$ becomes arbitrarily close to $1$. We say that "the limit of the sequence $n\cdot \sin\left\left(\tfrac1\right\right)$ equals $1$."
In
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 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 call ...
"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, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in ...
ultimately rests. Limits can be defined in any metric 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 point ...
, 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 is famous for formulating paradoxes that involve limiting processes. Leucippus,
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. N ...
,
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 prominen ...
, Eudoxus, and
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
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 area be ...
, 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 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 ...
s such as
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
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 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 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 re ...
, 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 call ...
$\left(x_n\right)$, 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, ...
$0.3333\dots$ 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 $\lim_ \left\left(1 + \tfrac\right\right)^n$ (the limit of which is the number ''e'') and the Arithmetic–geometric mean. The squeeze theorem 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 call ...
$\left(x_n\right)$, 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 ...
$N$ 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 $\left(x_n\right)$ is said to converge to or tend to the limit $x$. Symbolically, this is: :$\forall \varepsilon > 0 \left\left(\exists N \in \N \left\left(\forall n \in \N \left\left(n \geq N \implies , x_n - x, < \varepsilon \right\right)\right\right)\right\right)$. If a sequence $\left(x_n\right)$ converges to some limit $x$, then it is convergent and $x$ is the only limit; otherwise $\left(x_n\right)$ 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 $\left(a-\varepsilon,a+\varepsilon\right)$. 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 $\left(a-\varepsilon_1,a+\varepsilon_1\right)$. 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 ::$\lim_ \left(a_n \pm b_n\right) = \lim_ a_n \pm \lim_ b_n$ ::$\lim_ c a_n = c \cdot \lim_ a_n$ ::$\lim_ \left(a_n \cdot b_n\right) = \left\left(\lim_ a_n \right\right)\cdot \left\left( \lim_ b_n \right\right)$ ::$\lim_ \left\left(\frac\right\right) = \frac$ provided $\lim_ b_n \ne 0$ ::$\lim_ a_n^p = \left\left( \lim_ a_n \right\right)^p$ *For any
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
''f'', if $\lim_x_n$ exists, then $\lim_ f \left\left(x_n \right\right)$ exists too. In fact, any real-valued function ''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) 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) If $a_n$ is bounded and monotonic 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 $\left(x_n\right)$ 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: :$\forall K \in \mathbb \left\left(\exists N \in \N \left\left(\forall n \in \N \left\left(n \geq N \implies x_n > K \right\right)\right\right)\right\right)$. 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: :$\forall K \in \mathbb \left\left(\exists N \in \N \left\left(\forall n \in \N \left\left(n \geq N \implies x_n < K \right\right)\right\right)\right\right)$. 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 $x_n=\left(-1\right)^n$ 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 setti ...
$\left(X, d\right)$ 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 call ...
$\left(x_n\right)$ 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 ...
$N$ such that, for every natural number $n \geq N$, we have $d\left(x_n, x\right) < \varepsilon$. Symbolically, this is: :$\forall \varepsilon > 0 \left\left(\exists N \in \N \left\left(\forall n \in \N \left\left(n \geq N \implies d\left(x_n, x\right) < \varepsilon \right\right)\right\right)\right\right)$. This coincides with the definition given for real numbers when $X = \R$ and $d\left(x, y\right) = , x-y,$.

## 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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
''f'', if $\lim_ x_n$ exists, then $\lim_ f\left(x_n\right) = f\left\left(\lim_x_n \right\right)$. In fact, a function ''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, and, in particular, in real analysis. 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 spaces.

# Topological spaces

## Definition

A point $x \in X$ of the topological space $\left(X, \tau\right)$ 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 call ...
$\left\left(x_n\right\right)_$ if: :For every
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
$U$ 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 $\left(X, d\right)$ is a metric space and $\tau$ is the topology generated by $d$. A limit of a sequence of points $\left\left(x_n\right\right)_$ 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 of the affinely extended real number system, 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 cont ...
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 many ...
, 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, 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 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 $\left(x_n\right)$ 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 In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
). 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 $x_H$: :$L = \left(x_H\right)$. Thus, the limit can be defined by the formula :$\lim_ x_n= \left(x_H\right)$. 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 $\left(x_\right)$. 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 call ...
$\left(x_\right)$, 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 ...
$N$ 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 $\left(x_\right)$ is said to converge to or tend to the limit $x$. Symbolically, this is: :$\forall \varepsilon > 0 \left\left(\exists N \in \N \left\left(\forall n, m \in \N \left\left(n, m \geq N \implies , x_ - x, < \varepsilon \right\right)\right\right)\right\right)$. Note that the double limit is different from taking limit in ''n'' first, and then in ''m''. The latter is known as iterated limit. 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 $\left(x_\right)$ 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: :$\forall K \in \mathbb \left\left(\exists N \in \N \left\left(\forall n, m \in \N \left\left(n, m \geq N \implies x_ > K \right\right)\right\right)\right\right)$. Similarly, a sequence $\left(x_\right)$ 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: :$\forall K \in \mathbb \left\left(\exists N \in \N \left\left(\forall n, m \in \N \left\left(n, m \geq N \implies x_ < K \right\right)\right\right)\right\right)$. 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 $x_=\left(-1\right)^$ provides one such example.

## Pointwise limits and uniform limits

For a double sequence $\left(x_\right)$, we may take limit in one of the indices, say, $n \to \infty$, to obtain a single sequence $\left(y_m\right)$. 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 n ...
$m$, there exists a natural number $N\left(\varepsilon, m\right) > 0$ such that, for every natural number $n \geq N$, we have $, x_ - y_m, < \varepsilon$. Symbolically, this is: :$\forall \varepsilon > 0 \left\left( \forall m \in \mathbb \left\left(\exists N \in \N \left\left(\forall n \in \N \left\left(n \geq N \implies , x_ - y_m, < \varepsilon \right\right)\right\right)\right\right)\right\right)$. When such a limit exists, we say the sequence $\left(x_\right)$ converges pointwise to $\left(y_m\right)$. 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 $N\left(\varepsilon\right) > 0$ 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 n ...
$m$ and for every natural number $n \geq N$, we have $, x_ - y_m, < \varepsilon$. Symbolically, this is: :$\forall \varepsilon > 0 \left\left(\exists N \in \N \left\left( \forall m \in \mathbb \left\left(\forall n \in \N \left\left(n \geq N \implies , x_ - y_m, < \varepsilon \right\right)\right\right)\right\right)\right\right)$. 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 $\left(x_\right)$ converges uniformly to $\left(y_m\right)$.

## Iterated limit

For a double sequence $\left(x_\right)$, we may take limit in one of the indices, say, $n \to \infty$, to obtain a single sequence $\left(y_m\right)$, 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 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, which requires the limit $\lim_x_ = y_m$ to be uniform in ''m''.

*
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 cont ...
* Subsequential limit * Limit superior and limit inferior * Limit of a function * Limit of a sequence of functions * Limit of a sequence of sets * Limit of a net *
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 ...
*
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 ...
* Modes of convergence

# References

* * * Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow. * Frank Morley and James Harknessbr>A treatise on the theory of functions
(New York: Macmillan, 1893)