Limit of a sequence

TheInfoList

As the positive
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
$n$ 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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, the limit of a sequence is the value that the terms of a
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

"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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
ultimately rests. Limits can be defined in any
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
or
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, but are usually first encountered in the
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s.

# History

The Greek philosopher is famous for formulating paradoxes that involve limiting processes.
Leucippus Leucippus (; el, Λεύκιππος, ''Leúkippos''; fl. 5th century BCE) is reported in some ancient sources to have been a philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλό ...
,
Democritus Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient w ...

,
Antiphon An antiphon (Greek language, Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christianity, Christian ritual, sung as a refrain. The texts of antiphons are the Psalms. Their form was favored by St Ambrose a ...
, Eudoxus, and
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

developed the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface o ...

, 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 (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
.
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

s such as
Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

in his etude of
hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, speci ...
(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 Bernhard Bolzano (''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 mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

in the 1870s.

# Real numbers

In the
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

, a number $L$ is the limit of the
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$\left(x_n\right),$ 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 (mathematics), function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted \operatorname(x) or \lfloor x\rfloor ...
). For every $n \geq N,$ $, x_n - 0, \le x_N = \frac < \varepsilon.$
*If $x_n = 1/n$ 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 In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
$0.3333...$ 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The
squeeze theorem In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
is often useful in the establishment of such limits.

## Formal definition

We call $x$ the limit of the
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$\left(x_n\right)$ if the following condition holds: *For each
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
$\varepsilon > 0,$ there exists a
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
$N$ 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 $x,$ written $x_n \to x$ or $\lim_ x_n = x.$ 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 $\left(x_n\right)$ 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 $\epsilon_1 > 0$ 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

Limits of sequences behave well with respect to the usual
arithmetic operations Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...
. If $a_n \to a$ and $b_n \to b,$ then $a_n+b_n \to a+b,$ $a_n\cdot b_n \to ab$ and, if neither ''b'' nor any $b_n$ is zero, $\frac \to \frac.$ For any
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
''f'', if $x_n \to x$ then $f\left(x_n\right) \to f\left(x\right).$ In fact, any real-valued
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
''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). Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist). *The limit of a sequence is unique. *$\lim_ \left(a_n \pm b_n\right) = \lim_ a_n \pm \lim_ b_n$ *$\lim_ c a_n = c \cdot \lim_ a_n$ *$\lim_ \left(a_n \cdot b_n\right) = \left(\lim_ a_n\right)\cdot\left( \lim_ b_n\right)$ *$\lim_ \left\left(\frac\right\right) = \frac$ provided $\lim_ b_n \ne 0$ * *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 $n > N,$ and $\lim_ a_n = \lim_ b_n = L,$ then $\lim_ c_n = L.$ *If a sequence is Sequence#Bounded, bounded and Sequence#Increasing and decreasing, monotonic, 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 for every ''K'', there is an ''N'' such that for every $n \geq N,$ $x_n > K$; that is, the sequence terms are eventually larger than any fixed ''K''. Similarly, $x_n \to -\infty$ if for every ''K'', there is an ''N'' such that for every $n \geq N,$ $x_n < K.$ 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 $\left(X, d\right)$ is the limit of the
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$\left(x_n\right)$ if for all $\epsilon > 0,$ there is an $N$ such that, for every $n \geq N,$ $d\left(x_n, x\right) < \epsilon.$ This coincides with the definition given for real numbers when $X = \R$ and $d\left(x, y\right) = , x-y, .$

## Properties

For any
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
''f'', if $x_n \to x$ then $f\left(x_n\right) \to f\left(x\right).$ In fact, a Function (mathematics), function ''f'' is continuous if and only if it preserves the limits of sequences. Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for $\epsilon$ less than half this distance, sequence terms cannot be within a distance $\epsilon$ of both points.

# Topological spaces

## Definition

A point $x \in X$ of the topological space $\left(X, \tau\right)$ is a or of the
sequence In , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unl ...

$\left\left(x_n\right\right)_$ if for every Topological neighbourhood, neighbourhood $U$ of $x,$ there exists some $N \in \N$ such that for every $n \geq N,$ $x_n \in U.$ 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#Functions on topological spaces, limit of a function: the Domain of a function, domain is $\N$ in the space $\N \cup \lbrace + \infty \rbrace,$ with the induced topology of the affinely extended real number system, the Range of a function, range is $T,$ and the function argument $n$ tends to $+\infty,$ which in this space is a limit point of $\N.$

## Properties

In a Hausdorff space, 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.

# 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 space, complete metric spaces.

# Definition in 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). Equivalently, ''L'' is the Standard part function, standard part 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''.

* * * Limit superior and limit inferior * Modes of convergence * Net (mathematics)#Limits of nets, Limit of a net — A net (mathematics), net is a topological generalization of a sequence. * Set-theoretic limit * Shift rule *

# References

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