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1 Real numbers
2 In a metric space
3 Completeness
4 Generalizations
5 In constructive mathematics
6 See also
7 References

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-Louis 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, necessarily positive, value.

In other words, suppose a pre-assigned positive real value ε is chosen. However small ε 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 ε 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), the criterion for convergence 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 they 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.

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 rational or complex numbers.

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

$d(x_m, x_n) < \varepsilon$

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 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 (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 $x n = F n / F n - 1$ of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit $φ$ satisfying $φ 2 = φ + 1$ , 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.

Counter-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 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 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). 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 (x_n) 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

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.

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: A sequence $(x k)$ in a topological group $G$ is a Cauchy sequence if for every open neighbourhood $U$ of the identity in $G$ there 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.

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

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