Cauchy sequence

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.^[1] More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences.^[2]

It is not sufficient for each term to become arbitrarily close to the preceding 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 – their differences $a_{n+1}-a_{n}={\sqrt {n+1}}-{\sqrt {n}}={\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}<{\frac {1}{2{\sqrt {n}}}}$ tend to zero as the index $n$ grows. However, with growing values of $n$ , the terms $a_{n}$ become arbitrarily large. So, for any index $n$ and distance $d$ , there exists an index $m$ big enough such that $a_{m}-a_{n}>d.$ As a result, no matter how far one goes, the remaining terms of the sequence never get close to each other; 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 a metric space[edit]

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 $\left|x_{m}-x_{n}\right|$ 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.

The sequence defined by $x_{0}=1,x_{n+1}={\frac {x_{n}+2/x_{n}}{2}}$ consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational , see Babylonian method of computing square root.

square root of 2

The sequence $x_{n}=F_{n}/F_{n-1}$ of ratios of consecutive 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 $\varphi =(1+{\sqrt {5}})/2,$ the Golden ratio, which is irrational.

Fibonacci numbers

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

Generalizations[edit]

In topological vector spaces[edit]

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 each member $V\in B,$ 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[edit]

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 for every open neighbourhood $U$ of the identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x_{n}y_{m}^{-1}\in U.$ This relation is an equivalence relation: 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[edit]

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 (with respect to $H$ ) if and only if for any $r$ there is $N$ such that for all $m,n>N,x_{n}x_{m}^{-1}\in H_{r}.$

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 (sequences such that $\forall r,\exists N,\forall n>N,x_{n}\in H_{r}$ ) 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 (that is, 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 a hyperreal continuum[edit]

A real sequence $\langle u_{n}:n\in \mathbb {N} \rangle$ has a natural hyperreal extension, defined for hypernatural values H of the index n in addition to the usual natural n. The sequence is Cauchy if and only if for every infinite H and K, the values $u_{H}$ and $u_{K}$ are infinitely close, or adequal, that is,

– Annotated index of various modes of convergence

Modes of convergence (annotated index)

– Method of construction of the real numbers

Dedekind cut

(2012). Foundations of Constructive Analysis. Ishi Press. ISBN 9784871877145.

Bishop, Errett Albert

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]