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Encyclopedia > Cauchy sequence
The plot of a Cauchy sequence (xn), shown in blue, as n versus xn. If the space containing the sequence is complete, the "ultimate destination" of this sequence, that is, the limit, exists.
The plot of a Cauchy sequence (xn), shown in blue, as n versus xn. If the space containing the sequence is complete, the "ultimate destination" of this sequence, that is, the limit, exists.
A sequence that is not Cauchy. The elements of the sequence fail to get close to each other as the sequence progresses.
A sequence that is not Cauchy. The elements of the sequence fail to get close to each other as the sequence progresses.

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become 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 distances from any of the remaining elements to any other such element smaller than any preassigned positive value. Image File history File links Size of this preview: 800 × 560 pixelsFull resolution (2706 × 1894 pixel, file size: 76 KB, MIME type: image/png) % draw an illustration of a Cauchy sequence function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal; axis off; fontsize... Image File history File links Size of this preview: 800 × 560 pixelsFull resolution (2706 × 1894 pixel, file size: 76 KB, MIME type: image/png) % draw an illustration of a Cauchy sequence function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal; axis off; fontsize... Image File history File links Size of this preview: 800 × 512 pixelsFull resolution (2706 × 1733 pixel, file size: 80 KB, MIME type: image/png) % draw an illustration of a sequence that is not Cauchy function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal... Image File history File links Size of this preview: 800 × 512 pixelsFull resolution (2706 × 1733 pixel, file size: 80 KB, MIME type: image/png) % draw an illustration of a sequence that is not Cauchy function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... For other senses of this word, see sequence (disambiguation). ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...


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.


Because Cauchy sequences require the notion of distance, they can only be defined in a metric space. Their utility lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), 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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... The limit of a sequence is one of the oldest concepts in mathematical analysis. ...


The notions above are not as unfamiliar as 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 net. In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... In mathematics, a filter is a special subset of a partially ordered set. ... In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. ...

Contents

Cauchy sequence of real numbers

A sequence

x_1, x_2, x_3, ldots

of real numbers is called Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m,n > N A negative number is a number that is less than zero, such as −3. ... The integers are commonly denoted by the above symbol. ...

|x_m - x_n| < varepsilon,

where the vertical bars denote the absolute value. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...


In a similar way one can define Cauchy sequences of complex numbers.


Cauchy sequence in a metric space

To define Cauchy sequences in any metric space, the absolute value | xmxn | is replaced by the distance d(xm,xn) between xm and xn.


Formally, given a metric space (M, d), a sequence In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

x_1, x_2, x_3, ldots

is Cauchy, if for every positive real number ε > 0 there is a positive integer N such that for all natural numbers m,n > N, the distance In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The integers are commonly denoted by the above symbol. ...

d(xm,xn)

is less than ε. 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 M. Nonetheless, such a limit does not always exist within M. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...


Completeness

A metric space X in which every Cauchy sequence has a limit (in X) is called complete. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...


Examples

The real numbers are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, there are a number of ways of defining the real number system as an ordered field. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...


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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


Counter-example: rational numbers

The rational numbers Q are not complete (for the usual distance): There are sequences of rationals that converge (in R) to irrational numbers; 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: In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... 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. ...

  • 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 square root of two, see Babylonian method of computing square root.
  • The sequence xn = Fn / Fn − 1 of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit φ satisfying φ2 = φ + 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 phi = (1+sqrt5)/2, the Golden ratio, 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.

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. ... This article presents and explains several methods which can be used to calculate square roots. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ... Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ... As the degree of the taylor series rises, it approaches the correct function. ...

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 ( 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 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. 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 axiom. The alternative approach, mentioned above, of constructing the real numbers as the completionhttp://en.wikipedia.org/wiki/Completion_%28metric_space%29#Completion of the rational numbers, makes the completeness of the real numbers tautological. 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. ... The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ... In mathematical analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following are equivalent: S is closed and bounded every open cover of S has a finite subcover, that is, S is compact. ... The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...


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). Such a series  sum_{n=1}^{infty} x_{n} is considered to be convergent if and only if the sequence of partial sums (sm) is convergent, where  s_{m} = sum_{n=1}^{m} 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 , In mathematics, a series is a sum of a sequence of terms. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a series is a sum of a sequence of terms. ...

 s_{p} - s_{q} = sum_{n=q+1}^{p} x_{n}.

If is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then (f(xn)) is a Cauchy sequence in N. If (xn) and (yn) are two Cauchy sequences in the rational, real or complex numbers, then the sum (xn + yn) and the product (xnyn) are also Cauchy sequences. In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...


Generalizations

Cauchy sequences in topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0; then (xk) is a Cauchy sequence if for all members V of B, there is some number N such that whenever n,m > N,xnxm is an element of V. If the topology of X is compatible with a translation-invariant metric d, the two definitions agree. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...


Cauchy sequences in groups

There is also a concept of Cauchy sequence in a group G: Let H = (Hr) be a decreasing sequence of normal subgroups of G of finite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H) if and only if for any r there is N such that forall m,n > N, x_n x_m^{-1} in H_r. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... ↔ ⇔ ≡ logical symbols representing iff. ...


The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences (s.th. forall r, exists N, forall n > N, x_n in H_r) is a normal subgroup of C. The factor group C / C0 is called the completion of G with respect to H. In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...


One can then show that this completion is isomorphic to the inverse limit of the sequence (G / Hr). In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...


An example of this construction, familiar in number theory and algebraic geometry 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. Number 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. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...


If H is a cofinal sequence (i.e., any normal subgroup of finite index contains some Hr), then this completion is canonical in the sense that it is isomorphic to the inverse limit of (G / H)H, where H varies over all normal subgroups of finite index. For further details, see ch. I.10 in Lang's "Algebra". In mathematics, a subset B of a partially ordered set A is cofinal if for every a in A there is b in B such that a &#8804; b. ... Canonical is an adjective derived from canon. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...


In constructive mathematics

In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful. If (x1,x2,x3,...) is a Cauchy sequence in the set X, then a modulus of Cauchy convergence for the sequence is a function α from the set of natural numbers to itself, such that forall k forall m, n > alpha(k), |x_m - x_n| < 1/k. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...


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 α(k) be the smallest possible N in the definition of Cauchy sequence, taking r to be 1 / k). 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 AC00), 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. In mathematics, a well-order (or well-ordering) 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 law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ... In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. ...


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 α(k) = k or α(k) = 2k). 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 Bishop 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. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... Errett Albert Bishop (1928-1983) was an American mathematician known for is work on analysis. ...


References

  • Bourbaki, Nicolas (1972). Commutative Algebra, English translation, Addison-Wesley. ISBN 0-201-00644-8. 
  • Lang, Serge (1997). Algebra, 3rd ed., reprint w/ corr., Addison-Wesley. ISBN 978-0-201-55540-0. 
  • Spivak, Michael (1994). Calculus, 3rd ed., Berkeley, CA: Publish or Perish. ISBN 0-914098-89-6. 
  • Troelstra, A. S. and D. van Dalen. Constructivism in Mathematics: An Introduction.  (for uses in constructive mathematics)

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In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses.
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