Cauchy Sequence

In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

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:
$a_n=\sqrt n,$
the consecutive terms become arbitrarily close to each other:
$a_-a_n = \sqrt-\sqrt = \frac < \frac.$
However, with growing values of the index , the terms $a_n$ become arbitrarily large. So, for any index and distance , there exists an index big enough such that $a_m - a_n > d.$ (Actually, any $m > \left(\sqrt + d\right)^2$ suffices.) As a result, despite 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 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 algorithms, 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
$x_1, x_2, x_3, \ldots$
of real numbers is called a Cauchy sequence if for every positive real number $\varepsilon,$ 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 rational or complex numbers. Cauchy formulated such a condition by requiring $x_m - x_n$ 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 $r = \pi,$ this sequence is (3, 3.1, 3.14, 3.141, ...). The ''m''th and ''n''th terms differ by at most $10^$ when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number $\varepsilon.$

Modulus of Cauchy convergence

If $(x_1, x_2, x_3, ...)$ is a 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 for all natural numbers $k$ and natural numbers $m, n > \alpha(k),$ $, x_m - x_n, < 1/k.$

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 $\alpha(k)$ be the smallest possible $N$ in the definition of Cauchy sequence, taking $r$ to be $1/k$). The existence of a modulus also follows from the principle of dependent choice, which is a weak form of the axiom of choice, and it also follows from an even weaker condition called AC₀₀. ''Regular Cauchy sequences'' are 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 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 $\left, x_m - x_n\$ is replaced by the distance $d\left(x_m, x_n\right)$ (where ''d'' denotes a metric) between $x_m$ and $x_n.$

Formally, given a metric space $(X, d),$ a sequence
$x_1, x_2, x_3, \ldots$
is Cauchy, if for every positive real number $\varepsilon > 0$ there is a positive integer $N$ such that for all positive integers $m, n > N,$ the distance
$d\left(x_m, x_n\right) < \varepsilon.$

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 numbers 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 numbers. 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 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 a Cauchy sequence of rational numbers with irrational limit ''x''. Irrational numbers certainly exist in $\R,$ for example:

* The sequence defined by $x_0=1, x_=\frac$ 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 $x_n = F_n / F_$ 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 $\varphi = (1+\sqrt5)/2,$ 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 $x \neq 0,$ 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 $X = (0, 2)$ in the set of real numbers with an ordinary distance in $\R$ is not a complete space: there is a sequence $x_n = 1/n$ in it, which is Cauchy (for arbitrarily small distance bound $d > 0$ all terms $x_n$ 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 $\varepsilon > 0,$ beyond some fixed point, every term of the sequence is within distance $\varepsilon/2$ of ''s'', so any two terms of the sequence are within distance $\varepsilon$ of each other.
* In any metric space, a Cauchy sequence $x_n$ 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 $x_N$ and any terms up to the ''N''-th, then no term of the sequence has distance greater than $M + 1$ from $x_N$).
* 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