Cauchy sequence Totally Explained

Everything about Cauchy Sequence totally explained

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's 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.
   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.
   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 ť

x_1, x_2, x_3, ldots

of real numbers is called Cauchy, if for every positive real number ε > 0 there's a positive integer N such that for all natural numbers m,n > N ť

|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 ť

x_1, x_2, x_3, ldots

is Cauchy, if for every positive real number ε > 0 there's a positive integer N such that for all natural numbers m,n > N, the distance ť

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 doesn't 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 (x_n), 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, x_n+1 = (x_n + 2/x_n)/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_ in H_r

.
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 H_r is the additive subgroup consisting of integer multiples of p^r.
If

H

is a cofinal sequence (for example, any normal subgroup of finite index contains some

H_r

), then this completion is canonical in the sense that it's 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 doesn't 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'll 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) don't 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

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've 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 hasn't spread far outside of that milieu.

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