Cauchy sequence

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a 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 distances from any of the remaining elements to any other such element smaller than any preassigned positive value.

In other words, suppose a pre-assigned positive real value varepsilon is chosen. However small varepsilon 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 varepsilon 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 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.

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.

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, there is a positive integer N such that for all natural numbers m,n > N

|x_m - x_n| < varepsilon,

where the vertical bars denote the absolute value.

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 |x_m - x_n| is replaced by the distance d(x_m, x_n) between x_m and x_n.

Formally, given a metric space (M, d), a sequence

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

d(x_m, x_n)

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.

Completeness

A metric space X in which every Cauchy sequence has a limit (in X) is called complete.

Examples

The real numbers are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.

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.

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:

  • 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 x_n = F_n / F_{n-1} of ratios of consecutive Fibonacci numbers 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 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.

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 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). Such a series sum_{n=1}^{infty} x_{n} is considered to be convergent if and only if the sequence of partial sums (s_{m}) 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 ,

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

If f colon M rightarrow N is a uniformly continuous map between the metric spaces M and N and (xn) 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

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 (x_k) is a Cauchy sequence if for all members V of 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.

Cauchy sequences in topological groups

Since the topological vector space definition of Cauchy sequence only requires that there is a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence (x_k) in a topological group G is a Cauchy sequence if there for every open neighbourhood U of the identity in G exists some number N such that whenever m,n>N it follows that x_n x_m^{-1} 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 there for every open neighbourhood U of the identity in G exists some number N such that whenever m,n>N it follows that x_n y_m^{-1} in U. This relation is an equivalence relation. More precisely, it is reflexive since the sequences are Cauchy sequences. It is symmetric since y_n x_m^{-1} = (x_m y_n^{-1})^{-1} in U^{-1} which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since x_n z_l^{-1} = x_n y_m^{-1} y_m z_l^{-1} 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.

Cauchy sequences in groups

There is also a concept of Cauchy sequence in a group G: Let H=(H_r) be a decreasing sequence of normal subgroups of G of finite index. Then a sequence (x_n) 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.

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 (s.th. 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 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.

If H is a cofinal sequence (i.e., 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 all normal subgroups of finite index. For further details, see ch. I.10 in Lang's "Algebra".

In constructive mathematics

In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful. If (x_1, x_2, x_3, ...) is a Cauchy sequence in the set X, then a modulus of Cauchy convergence for the sequence is a function alpha from the set of natural numbers to itself, such that forall k forall m, n > alpha(k), |x_m - x_n| < 1/k.

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 alpha(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.

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 alpha(k) = k or alpha(k) = 2^k). 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.

See also

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.; D. van Dalen Constructivism in Mathematics: An Introduction. (for uses in constructive mathematics)

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