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:
the consecutive terms become arbitrarily close to each other – their differences
tend to zero as the index grows. However, with growing values of , the terms
become arbitrarily large. So, for any index and distance , there exists an index big enough such that
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
of real numbers is called a Cauchy sequence if for every
positive real number
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 ...
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
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
this sequence is (3, 3.1, 3.14, 3.141, ...). The ''m''th and ''n''th terms differ by at most
when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number
Modulus of Cauchy convergence
If
is a sequence in the set
then a ''modulus of Cauchy convergence'' for the sequence is a
function 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
and natural numbers
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
be the smallest possible
in the definition of Cauchy sequence, taking
to be
). 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
or
). 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
is replaced by the distance
(where ''d'' denotes a
metric) between
and
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 ...
a sequence of elements of
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 ...
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 ...
such that for all positive integers
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
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
are not complete (for the usual distance):
There are sequences of rationals that converge (in
) 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
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
for example:
* The sequence defined by
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
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
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 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
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
in the set of real numbers with an ordinary distance in
is not a complete space: there is a sequence
in it, which is Cauchy (for arbitrarily small distance bound
all terms
of
fit in the
interval), however does not converge in
— its 'limit', number 0, does not belong to the space
Other properties
* Every convergent sequence (with limit ''s'', say) is a Cauchy sequence, since, given any real number
beyond some fixed point, every term of the sequence is within distance
of ''s'', so any two terms of the sequence are within distance
of each other.
* In any metric space, a Cauchy sequence
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
and any terms up to the ''N''-th, then no term of the sequence has distance greater than
from
).
* 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
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
If
is a
uniformly continuous map between the metric spaces ''M'' and ''N'' and (''x''
''n'') 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.
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 ...
: Pick a
local base for
about 0; then (
) is a Cauchy sequence if for each member
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 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
in a topological group
is a Cauchy sequence if for every open neighbourhood
of the
identity 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 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 ...
of the identity in
there exists some number
such that whenever
it follows that
This relation is an
equivalence relation: 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 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
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
in
is said to be Cauchy (with respect to
) if and only if for any
there is
such that for all
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 (sequences such that
) 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 limit of the sequence
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
-adic completion of the integers with respect to a
prime In this case,
is the integers under addition, and
is the additive subgroup consisting of integer multiples of
If
is a
cofinal sequence (that is, any normal subgroup of finite index contains some
), then this completion is
canonical in the sense that it is isomorphic to the inverse limit of
where
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
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
and
are infinitely close, or
adequal, that is,
:
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
(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
), this Cauchy completion yields
(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)