In

_{00}. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually $\backslash alpha(k)\; =\; k$ or $\backslash alpha(k)\; =\; 2^k$). 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.

There are sequences of rationals that converge (in $\backslash R$) to_{''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 $\backslash R,$ for example:
* The sequence defined by $x\_0=1,\; x\_=\backslash frac$ consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the

_{''n''}) is a Cauchy sequence in ''M'', then $(f(x\_n))$ is a Cauchy sequence in ''N''. If $(x\_n)$ and $(y\_n)$ are two Cauchy sequences in the rational, real or complex numbers, then the sum $(x\_n\; +\; y\_n)$ and the product $(x\_n\; y\_n)$ are also Cauchy sequences.

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 ...

, 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 calle ...

whose elements 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:
$$a\_n=\backslash sqrt\; n,$$
the consecutive terms become arbitrarily close to each other:
$$a\_-a\_n\; =\; \backslash sqrt-\backslash sqrt\; =\; \backslash frac\; <\; \backslash frac.$$
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 $m\; >\; \backslash left(\backslash sqrt\; +\; d\backslash right)^2$ 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
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four We ...

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 exist in the form of Cauchy filters and Cauchy nets.
In real numbers

A sequence $$x\_1,\; x\_2,\; x\_3,\; \backslash ldots$$ of real numbers is called a Cauchy sequence if for every positive real number $\backslash varepsilon,$ there is a positiveinteger
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 ...

''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 ...

$m,\; n\; >\; N,$
$$,\; x\_m\; -\; x\_n,\; <\; \backslash varepsilon,$$
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), a ...

. 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 refer ...

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 $r\; =\; \backslash pi,$ 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 $\backslash varepsilon.$
Modulus of Cauchy convergence

If $(x\_1,\; x\_2,\; x\_3,\; ...)$ is a sequence in the set $X,$ then a ''modulus of Cauchy convergence'' for the sequence is afunction
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-orient ...

$\backslash alpha$ 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 $m,\; n\; >\; \backslash alpha(k),$ $,\; x\_m\; -\; x\_n,\; <\; 1/k.$
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 of the natural numbers (let $\backslash alpha(k)$ 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, which is a weak form of the axiom of choice, and it also follows from an even weaker condition called ACIn 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 $\backslash left,\; x\_m\; -\; x\_n\backslash $ is replaced by the distance $d\backslash left(x\_m,\; x\_n\backslash right)$ (where ''d'' denotes ametric
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 setti ...

$(X,\; d),$ a sequence
$$x\_1,\; x\_2,\; x\_3,\; \backslash ldots$$
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. Every ...

$\backslash varepsilon\; >\; 0$ 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 language ...

$N$ such that for all positive integers $m,\; n\; >\; N,$ the distance
$$d\backslash left(x\_m,\; x\_n\backslash right)\; <\; \backslash varepsilon.$$
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 calledcomplete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...

.
Examples

Thereal 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 are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers 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 g ...

(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

Therational 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 $\backslash Q$ are not complete (for the usual distance):There are sequences of rationals that converge (in $\backslash R$) 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 $\backslash Q.$ In fact, if a real number ''x'' is irrational, then the sequence (''x''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 $\backslash phi$ satisfying $\backslash phi^2\; =\; \backslash phi+1,$ and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number $\backslash varphi\; =\; (1+\backslash sqrt5)/2,$ 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 $x\; \backslash neq\; 0,$ 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 MacLauri ...

.
Non-example: open interval

The open interval $X\; =\; (0,\; 2)$ in the set of real numbers with an ordinary distance in $\backslash R$ 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 $(0,\; d)$ 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 $\backslash varepsilon\; >\; 0,$ beyond some fixed point, every term of the sequence is within distance $\backslash varepsilon/2$ of ''s'', so any two terms of the sequence are within distance $\backslash varepsilon$ 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, 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. 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, orBanach 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 vecto ...

). Such a series
$\backslash sum\_^\; x\_n$ is considered to be convergent if and only if the sequence of partial sums $(s\_)$ is convergent, where $s\_m\; =\; \backslash sum\_^\; x\_n.$ It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers $p\; >\; q,$
$$s\_p\; -\; s\_q\; =\; \backslash sum\_^p\; x\_n.$$
If $f\; :\; M\; \backslash to\; N$ 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''Generalizations

In topological vector spaces

There is also a concept of Cauchy sequence for atopological 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 $B$ for $X$ about 0; then ($x\_k$) is a Cauchy sequence if for each member $V\backslash in\; B,$ 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 atopological 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 s ...

: A sequence $(x\_k)$ 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 $x\_n\; x\_m^\; \backslash in\; U.$ 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 $(x\_k)$ and $(y\_k)$ 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 ...

$U$ of the identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x\_n\; y\_m^\; \backslash in\; U.$ 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 relation ...

: It is reflexive since the sequences are Cauchy sequences. It is symmetric since $y\_n\; x\_m^\; =\; (x\_m\; y\_n^)^\; \backslash in\; U^$ which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since $x\_n\; z\_l^\; =\; x\_n\; y\_m^\; y\_m\; z\_l^\; \backslash in\; U\text{'}\; U\text{'}\text{'}$ where $U\text{'}$ and $U\text{'}\text{'}$ are open neighbourhoods of the identity such that $U\text{'}U\text{'}\text{'}\; \backslash subseteq\; U$; 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 $H=(H\_r)$ be a decreasing sequence ofnormal 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 $(x\_n)$ 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 $m,\; n\; >\; N,\; x\_n\; x\_m^\; \backslash in\; H\_r.$
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 $\backslash forall\; r,\; \backslash exists\; N,\; \backslash forall\; n\; >\; N,\; x\_n\; \backslash in\; H\_r$) 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 $(G/H\_r).$
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, "Math ...

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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

$p.$ 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
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...

in the sense that it is isomorphic to the inverse limit of $(G/H)\_H,$ 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 $\backslash langle\; u\_n\; :\; n\; \backslash in\; \backslash N\; \backslash rangle$ has a naturalhyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality
Described by Jean Baudrillar ...

extension, defined for hypernatural 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,
:$\backslash mathrm(u\_H-u\_K)=\; 0$
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 s ...

.
Cauchy completion of categories

introduced a notion of Cauchy completion of acategory
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 $\backslash Q$ (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 a ...

from ''x'' to ''y'' if and only if $x\; \backslash leq\; y$), this Cauchy completion yields $\backslash R\backslash cup\backslash left\backslash $ (again interpreted as a category using its natural ordering).
See also

* *References

Further reading

* * * * * * * (for uses in constructive mathematics)External links

* {{series (mathematics) Augustin-Louis Cauchy Metric geometry Topology Abstract algebra Sequences and series Convergence (mathematics)