Cauchy sequence

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In mathematics, a Cauchy sequence (; ), named after
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, is 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 ...
whose
elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...
become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: $a_n=\sqrt n,$ the consecutive terms become arbitrarily close to each other: $a_-a_n = \sqrt-\sqrt = \frac < \frac.$ However, with growing values of the index , the terms $a_n$ become arbitrarily large. So, for any index and distance , there exists an index big enough such that $a_m - a_n > d.$ (Actually, any $m > \left\left(\sqrt + d\right\right)^2$ suffices.) As a result, despite how far one goes, the remaining terms of the sequence never get close to ; hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a
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 ...
(one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Generalizations of Cauchy sequences in more abstract
uniform spaces In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
exist in the form of
Cauchy filter In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uni ...
s and
Cauchy 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 codomai ...
s.

# In real numbers

A sequence $x_1, x_2, x_3, \ldots$ of real numbers is called a Cauchy sequence if for every positive real number $\varepsilon,$ there is a 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'' such that for all
natural numbers 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 ...
$m, n > N,$ $, x_m - x_n, < \varepsilon,$ where the vertical bars denote the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring $x_m - x_n$ to be
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 ref ...
for every pair of infinite ''m'', ''n''. For any real number ''r'', the sequence of truncated decimal expansions of ''r'' forms a Cauchy sequence. For example, when $r = \pi,$ this sequence is (3, 3.1, 3.14, 3.141, ...). The ''m''th and ''n''th terms differ by at most $10^$ when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number $\varepsilon.$

## Modulus of Cauchy convergence

If $\left(x_1, x_2, x_3, ...\right)$ is a sequence in the set $X,$ then a ''modulus of Cauchy convergence'' for the sequence is a function $\alpha$ from 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 ...
s to itself, such that for all natural numbers $k$ and natural numbers $m, n > \alpha\left(k\right),$ $, x_m - x_n, < 1/k.$ Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the
well-ordering property In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
of the natural numbers (let $\alpha\left(k\right)$ be the smallest possible $N$ in the definition of Cauchy sequence, taking $r$ to be $1/k$). The existence of a modulus also follows from the principle of
dependent choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores whi ...
, which is a weak form of the axiom of choice, and it also follows from an even weaker condition called AC00. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually $\alpha\left(k\right) = k$ or $\alpha\left(k\right) = 2^k$). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice. Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by and by in constructive mathematics textbooks.

# In a metric space

Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space ''X''. To do so, the absolute value $\left, x_m - x_n\$ is replaced by the distance $d\left\left(x_m, x_n\right\right)$ (where ''d'' denotes a
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 mathema ...
) between $x_m$ and $x_n.$ Formally, given 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 set ...
$\left(X, d\right),$ a sequence $x_1, x_2, x_3, \ldots$ is Cauchy, if for every positive
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. Ever ...
$\varepsilon > 0$ there is a 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 ...
$N$ such that for all positive integers $m, n > N,$ the distance $d\left(x_m, x_n\right) < \varepsilon.$ Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in ''X''. Nonetheless, such a limit does not always exist within ''X'': the property of a space that every Cauchy sequence converges in the space is called ''completeness'', and is detailed below.

# Completeness

A metric space (''X'', ''d'') in which every Cauchy sequence converges to an element of ''X'' is called complete.

## Examples

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. Ever ...
s are complete under the metric induced by the usual absolute value, and one of the standard
constructions of the real numbers In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
involves Cauchy sequences 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 ratio ...
s. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number. A rather different type of example is afforded by a metric space ''X'' which has 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 ...
(where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of ''X'' must be constant beyond some fixed point, and converges to the eventually repeating term.

## Non-example: rational numbers

The
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 ratio ...
s $\Q$ are not complete (for the usual distance):
There are sequences of rationals that converge (in $\R$) to
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s; these are Cauchy sequences having no limit in $\Q.$ In fact, if a real number ''x'' is irrational, then the sequence (''x''''n''), whose ''n''-th term is the truncation to ''n'' decimal places of the decimal expansion of ''x'', gives a Cauchy sequence of rational numbers with irrational limit ''x''. Irrational numbers certainly exist in $\R,$ for example: * The sequence defined by $x_0=1, x_=\frac$ consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the
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 ...
square root of two, see Babylonian method of computing square root. * The sequence $x_n = F_n / F_$ of ratios of consecutive
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s which, if it converges at all, converges to a limit $\phi$ satisfying $\phi^2 = \phi+1,$ and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number $\varphi = \left(1+\sqrt5\right)/2,$ the
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, which is irrational. * The values of the exponential, sine and cosine functions, exp(''x''), sin(''x''), cos(''x''), are known to be irrational for any rational value of $x \neq 0,$ but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
.

## Non-example: open interval

The open interval $X = \left(0, 2\right)$ in the set of real numbers with an ordinary distance in $\R$ is not a complete space: there is a sequence $x_n = 1/n$ in it, which is Cauchy (for arbitrarily small distance bound $d > 0$ all terms $x_n$ of $n > 1/d$ fit in the $\left(0, d\right)$ interval), however does not converge in $X$ — its 'limit', number 0, does not belong to the space $X .$

## Other properties

* Every convergent sequence (with limit ''s'', say) is a Cauchy sequence, since, given any real number $\varepsilon > 0,$ beyond some fixed point, every term of the sequence is within distance $\varepsilon/2$ of ''s'', so any two terms of the sequence are within distance $\varepsilon$ of each other. * In any metric space, a Cauchy sequence $x_n$ is bounded (since for some ''N'', all terms of the sequence from the ''N''-th onwards are within distance 1 of each other, and if ''M'' is the largest distance between $x_N$ and any terms up to the ''N''-th, then no term of the sequence has distance greater than $M + 1$ from $x_N$). * In any metric space, a Cauchy sequence which has a convergent subsequence with limit ''s'' is itself convergent (with the same limit), since, given any real number ''r'' > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance ''r''/2 of ''s'', and any two terms of the original sequence are within distance ''r''/2 of each other, so every term of the original sequence is within distance ''r'' of ''s''. These last two properties, together with the
Bolzano–Weierstrass theorem In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that e ...
, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the
least upper bound axiom In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
. The alternative approach, mentioned above, of the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete
normed linear space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
, or
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
). Such a series $\sum_^ x_n$ is considered to be convergent if and only if the sequence of
partial sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s $\left(s_\right)$ is convergent, where $s_m = \sum_^ x_n.$ It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers $p > q,$ $s_p - s_q = \sum_^p x_n.$ If $f : M \to N$ is a
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
map between the metric spaces ''M'' and ''N'' and (''x''''n'') is a Cauchy sequence in ''M'', then $\left(f\left(x_n\right)\right)$ is a Cauchy sequence in ''N''. If $\left(x_n\right)$ and $\left(y_n\right)$ are two Cauchy sequences in the rational, real or complex numbers, then the sum $\left(x_n + y_n\right)$ and the product $\left(x_n y_n\right)$ are also Cauchy sequences.

# Generalizations

## In topological vector spaces

There is also a concept of Cauchy sequence for a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
$X$: Pick a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
$B$ for $X$ about 0; then ($x_k$) is a Cauchy sequence if for each member $V\in B,$ there is some number $N$ such that whenever $n,m > N, x_n - x_m$ is an element of $V.$ If the topology of $X$ is compatible with a translation-invariant metric $d,$ the two definitions agree.

## In topological groups

Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
: A sequence $\left(x_k\right)$ in a topological group $G$ is a Cauchy sequence if for every open neighbourhood $U$ of the identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x_n x_m^ \in U.$ As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in $G.$ As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in $G$ that $\left(x_k\right)$ and $\left(y_k\right)$ are equivalent if for every open
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 the identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x_n y_m^ \in U.$ This relation is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
: It is reflexive since the sequences are Cauchy sequences. It is symmetric since $y_n x_m^ = \left(x_m y_n^\right)^ \in U^$ which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since $x_n z_l^ = x_n y_m^ y_m z_l^ \in U\text{'} U\text{'}\text{'}$ where $U\text{'}$ and $U\text{'}\text{'}$ are open neighbourhoods of the identity such that $U\text{'}U\text{'}\text{'} \subseteq U$; such pairs exist by the continuity of the group operation.

## In groups

There is also a concept of Cauchy sequence in a group $G$: Let $H=\left(H_r\right)$ be a decreasing sequence of
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
s of $G$ of finite
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
. Then a sequence $\left(x_n\right)$ in $G$ is said to be Cauchy (with respect to $H$) if and only if for any $r$ there is $N$ such that for all $m, n > N, x_n x_m^ \in H_r.$ Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on $G,$ namely that for which $H$ is a local base. The set $C$ of such Cauchy sequences forms a group (for the componentwise product), and the set $C_0$ of null sequences (sequences such that $\forall r, \exists N, \forall n > N, x_n \in H_r$) is a normal subgroup of $C.$ The
factor group Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, s ...
$C/C_0$ is called the completion of $G$ with respect to $H.$ One can then show that this completion is isomorphic to the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits c ...
of the sequence $\left(G/H_r\right).$ An example of this construction familiar in
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
is the construction of the $p$-adic completion of the integers with respect to a prime $p.$ In this case, $G$ is the integers under addition, and $H_r$ is the additive subgroup consisting of integer multiples of $p_r.$ If $H$ is a cofinal sequence (that is, any normal subgroup of finite index contains some $H_r$), then this completion is canonical in the sense that it is isomorphic to the inverse limit of $\left(G/H\right)_H,$ where $H$ varies over normal subgroups of finite
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
. For further details, see Ch. I.10 in Lang's "Algebra".

## In a hyperreal continuum

A real sequence $\langle u_n : n \in \N \rangle$ has a natural hyperreal extension, defined for
hypernatural In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
values ''H'' of the index ''n'' in addition to the usual natural ''n''. The sequence is Cauchy if and only if for every infinite ''H'' and ''K'', the values $u_H$ and $u_K$ are infinitely close, or adequal, that is, :$\mathrm\left(u_H-u_K\right)= 0$ where "st" is the
standard part function 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 ...
.

## Cauchy completion of categories

introduced a notion of Cauchy completion of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. Applied to $\Q$ (the category whose objects are rational numbers, and there is a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
from ''x'' to ''y'' if and only if $x \leq y$), this Cauchy completion yields $\R\cup\left\$ (again interpreted as a category using its natural ordering).