Cauchy sequence
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Cauchy sequence 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 cal ...
whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
; they may occasionally be known as fundamental sequences. 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 – their differences a_-a_n = \sqrt-\sqrt = \frac < \frac tend to zero as the index grows. However, with growing values of , 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. As a result, no matter 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 (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), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
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 Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
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, \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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''N'' such that for all
natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. In a similar way one can define Cauchy sequences of rational or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. Cauchy formulated such a condition by requiring x_m - x_n to be infinitesimal 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 (x_1, x_2, x_3, ...) 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s to itself, such that for all natural numbers k and natural numbers m, n > \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 \alpha(k) be the smallest possible N in the definition of Cauchy sequence, taking \varepsilon to be 1/k). The existence of a modulus also follows from the principle of countable choice. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually \alpha(k) = k or \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.


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(x_m, x_n\right) (where ''d'' denotes a metric) between x_m and x_n. Formally, given a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
(X, d), a sequence of elements of X 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
\varepsilon > 0 there is a positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 (for example, The set of all ...
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 (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 (for example, The set of all ...
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 are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
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
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, see Babylonian method of computing square root. * The sequence x_n = F_n / F_ of ratios of consecutive
Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
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 = (1+\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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
, 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.


Non-example: open interval

The open interval X = (0, 2) 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 (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 \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, 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, or
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
). Such a series \sum_^ x_n is considered to be convergent if and only if the sequence of partial sums (s_) 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 map between the metric spaces ''M'' and ''N'' and (''x''''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.


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 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 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 structures ...
: 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^ \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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
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: It is reflexive since the sequences are Cauchy sequences. It is symmetric since y_n x_m^ = (x_m y_n^)^ \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' U'' where U' and U'' are open neighbourhoods of the identity such that U'U'' \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 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 ...
s of G of finite
index Index (: indexes or 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 the Halo Array in the ...
. 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^ \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 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 of the sequence (G/H_r). An example of this construction familiar in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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 (G/H)_H, where H varies over normal subgroups of finite
index Index (: indexes or 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 the Halo Array in the ...
. 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 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(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 suc ...
.


Cauchy completion of categories

introduced a notion of Cauchy completion of a category. Applied to \Q (the category whose objects are rational numbers, and there is a
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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).


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)