In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
Cauchy sequence, named after Augustin Cauchy, is a
sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence...
whose elements become
arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the
distancesIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
from any of the remaining elements to any other such element smaller than any preassigned, necessarily positive, value.
In other words, suppose a pre-assigned positive real value is chosen.
However small is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will be within distance of each other.
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 limitThe limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit.Intuitively, suppose we have a sequence of points The limit of a sequence is one of the oldest concepts in...
), they give a criterion for convergence which depends only on the terms of the sequence itself. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates.
The notions above are not as unfamiliar as they might at first appear. The customary acceptance of the fact that any real number
x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of
x) has the real limit
x. In some cases it may be difficult to describe
x independently of such a limiting process involving rational numbers.
Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filter and
Cauchy netIn mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.A net is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, is a member of V...
.
Real numbers
A sequence
of real numbers is called
Cauchy, if for every positive real number
ε, there is a positive
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
N such that for all natural numbers
m,
n >
N
where the vertical bars denote the
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
.
In a similar way one can define Cauchy sequences of complex numbers.
In a metric space
To define Cauchy sequences in any metric space, the absolute value is replaced by the
distance between and .
Formally, given a
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(
M,
d), a sequence
is Cauchy, if for every positive
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
ε > 0 there is a positive
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
N such that for all natural numbers
m,
n >
N, the distance
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
limitThe limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit.Intuitively, suppose we have a sequence of points The limit of a sequence is one of the oldest concepts in...
in
M. Nonetheless, such a limit does not always exist within
M.
Completeness
A metric space
X in which every Cauchy sequence has a limit in
X is called complete.
Examples
The
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s.
A rather different type of example is afforded by a metric space
X which has the
discrete metricIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(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.
Counter-example: rational numbers
The
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s
Q are not complete (for the usual distance):
There are sequences of rationals that converge (in
R) to
irrational numberIn mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
s; these are Cauchy sequences having no limit in
Q. In fact, if a real number
x is irrational, then the sequence (
xn), whose
n-th term is the truncation to
n decimal places of the decimal expansion of
x, gives Cauchy sequence of rational numbers with irrational limit
x. Irrational numbers certainly exist, for example:
- The sequence defined by x0 = 1, xn+1 = (xn + 2/xn)/2 consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational
In mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
square root of two, see Babylonian method of computing square root.
- The sequence of ratios of consecutive Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following sequence:By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two...
s which, if it converges at all, converges to a limit satisfying , and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number , the Golden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887...
, 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≠0, but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.
Counter-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
xn=1/
n in it, which is Cauchy (for arbitrarily small distance bound
d>0 all terms
xn 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 r > 0, beyond some fixed point, every term of sequence is within distance r/2 of s, so any two terms of the sequence are within distance r of each other.
- Every Cauchy sequence of real (or complex) numbers is bounded
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M < ∞ such thatfor all x in X....
(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 absolute value of the terms up to and including the N-th, then no term of the sequence has absolute value greater than M+1).
- 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 a lemma used in the proof of the
Bolzano–Weierstrass theoremIn real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the
Heine–Borel theoremIn the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...
. The lemma in question states that every bounded sequence of real numbers has a convergent subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has a convergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the
least upper bound axiomThe least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis stating that if a nonempty subset of the real numbers has an upper bound, then it has a least upper bound. It is an axiom in the sense that it cannot be proven by the other axioms within the system of...
. The alternative approach, mentioned above, of
constructing the real numbers as the 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 spaceIn mathematics, Banach spaces are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions , spaces of Lebesgue integrable functions known as Lp spaces,...
). Such a series
is considered to be convergent if and only if the sequence of partial sums is convergent, where
. It is a routine matter
to determine whether the sequence of partial sums is Cauchy or not,
since for positive integers
p >
q,
.
If is a uniformly continuous map between the metric spaces
M and
N and (
xn) is a Cauchy sequence in
M, then is a Cauchy sequence in
N. If and are two Cauchy sequences in the rational, real or complex numbers, then the sum and the product are also Cauchy sequences.
In topological vector spaces
There is also a concept of Cauchy sequence for a
topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
: Pick a local base for about 0; then is a Cauchy sequence if for all members of , there is some number such that whenever
is an element of . If the topology of is compatible with a translation-invariant metric , 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 groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology...
: A sequence in a topological group is a Cauchy sequence if for every open neighbourhood of the
identityIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
in there exists some number such that whenever it follows that . As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in .
As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in that and are
equivalent if there for every open neighbourhood of the identity in exists some number such that whenever it follows that . This relation is an
equivalence relationIn mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets...
. More precisely, it is reflexive since the sequences are Cauchy sequences. It is symmetric since which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since where and are open neighbourhoods of the identity such that ; such pairs exist by the continuity of the group operation.
In groups
There is also a concept of Cauchy sequence in a group :
Let be a decreasing sequence of normal subgroups of of finite
indexIn mathematics, specifically group theory, the index of a subgroup H in a group G is the “relative size” of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively “half” of the elements of G lie in H...
.
Then a sequence in is said to be Cauchy (w.r.t. )
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
for any there is such that .
Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on , namely that for which is a local base.
The set of such Cauchy sequences forms a group (for the componentwise product), and the set of null sequences (s.th. ) is a normal subgroup of . The factor group is called the completion of with respect to .
One can then show that this completion is isomorphic to the
inverse limitIn mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
of the sequence .
An example of this construction, familiar in
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
and
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
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
Hr is the additive subgroup consisting of integer multiples of
pr.
If is a
cofinalIn mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:This definition is most commonly applied when B is a partially ordered set or directed set under the relation ≤. Also, the notion of cofinal...
sequence (i.e., any normal subgroup of finite index contains some ), then this completion is
canonicalCanonical is an adjective derived from canon. Canon comes from the Greek word kanon, "rule" , and is used in various meanings....
in the sense that it is isomorphic to the inverse limit of , where varies over
all normal subgroups of finite
indexIn mathematics, specifically group theory, the index of a subgroup H in a group G is the “relative size” of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively “half” of the elements of G lie in H...
.
For further details, see ch. I.10 in
LangSerge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group....
's "Algebra".
In constructive mathematics
In constructive mathematics, Cauchy sequences often must be given with a
modulus of Cauchy convergence to be useful. If is a Cauchy sequence in the set , then a modulus of Cauchy convergence for the sequence is a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
from the set of
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s to itself, such that .
Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchy sequence has a modulus) follows from the well-ordering property of the natural numbers (let be the smallest possible in the definition of Cauchy sequence, taking to be ). However, this well-ordering property does not hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, this converse also follows (directly) from the principle of dependent choice (in fact, it will follow from the weaker AC
00), which is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directly only by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.
That said, using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Perhaps even more useful are
regular Cauchy sequences, sequences with a given modulus of Cauchy convergence (usually or ). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice. Regular Cauchy sequences were used by
Errett BishopErrett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructivist analysis, by virtue of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
in his
Foundations of Constructive Analysis, but they have also been used by Douglas Bridges in a non-constructive textbook (ISBN 978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread far outside of that milieu.