Cauchy sequence

If ${\displaystyle (x_{1},x_{2},x_{3},...)}$ is a sequence in the set ${\displaystyle X,}$ then a modulus of Cauchy convergence for the sequence is a function ${\displaystyle \alpha }$ from the set of natural numbers to itself, such that for all natural numbers ${\displaystyle k}$ and natural numbers ${\displaystyle m,n>\alpha (k),}$ ${\displaystyle |x_{m}-x_{n}|<1/k.}$

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 ${\displaystyle \alpha (k)}$ be the smallest possible ${\displaystyle N}$ in the definition of Cauchy sequence, taking ${\displaystyle r}$ to be ${\displaystyle 1/k}$). 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 ${\displaystyle \alpha (k)=k}$ or ${\displaystyle \alpha (k)=2^{k}}$). 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 Bishop (2012) and by Bridges (1997) in constructive mathematics textbooks.

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.

The rational numbers ${\displaystyle \mathbb {Q} }$ are not complete (for the usual distance):
There are sequences of rationals that converge (in ${\displaystyle \mathbb {R} }$) to irrational numbers; these are Cauchy sequences having no limit in ${\displaystyle \mathbb {Q} .}$ 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 ${\displaystyle \mathbb {R} ,}$ for example:

• The sequence defined by ${\displaystyle x_{0}=1,x_{n+1}={\frac {x_{n}+{\frac {2}{x_{n}}}}{2}}}$ 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 ${\displaystyle x_{n}=F_{n}/F_{n-1}}$ of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit ${\displaystyle \phi }$ satisfying ${\displaystyle \phi ^{2}=\phi +1,}$ and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number ${\displaystyle \varphi =(1+{\sqrt {5}})/2,}$ 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 ${\displaystyle x\neq 0,}$ but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.

The open interval ${\displaystyle X=(0,2)}$ in the set of real numbers with an ordinary distance in ${\displaystyle \mathbb {R} }$ is not a complete space: there is a sequence ${\displaystyle x_{n}=1/n}$ in it, which is Cauchy (for arbitrarily small distance bound ${\displaystyle d>0}$ all terms ${\displaystyle x_{n}}$ of ${\displaystyle n>1/d}$ fit in the ${\displaystyle (0,d)}$ interval), however does not converge in ${\displaystyle X}$ — its 'limit', number 0, does not belong to the space ${\displaystyle X.}$

• Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ${\displaystyle \varepsilon >0,}$ beyond some fixed point, every term of the sequence is within distance ${\displaystyle \varepsilon /2}$ of s, so any two terms of the sequence are within distance ${\displaystyle \varepsilon }$ of each other.
• In any metric space, a Cauchy sequence ${\displaystyle x_{n}}$ 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 ${\displaystyle x_{N}}$ and any terms up to the N-th, then no term of the sequence has distance greater than ${\displaystyle M+1}$ from ${\displaystyle x_{N}}$).
• 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 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 ${\textstyle \sum _{n=1}^{\infty }x_{n}}$ is considered to be convergent if and only if the sequence of partial sums ${\displaystyle (s_{m})}$ is convergent, where ${\textstyle s_{m}=\sum _{n=1}^{m}x_{n}.}$ It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers ${\displaystyle p>q,}$

$${\displaystyle s_{p}-s_{q}=\sum _{n=q+1}^{p}x_{n}.}$$

If ${\displaystyle f:M\to N}$ is a uniformly continuous map between the metric spaces M and N and (x_n) is a Cauchy sequence in M, then ${\displaystyle (f(x_{n}))}$ is a Cauchy sequence in N. If ${\displaystyle (x_{n})}$ and ${\displaystyle (y_{n})}$ are two Cauchy sequences in the rational, real or complex numbers, then the sum ${\displaystyle (x_{n}+y_{n})}$ and the product ${\displaystyle (x_{n}y_{n})}$ are also Cauchy sequences.

There is also a concept of Cauchy sequence for a topological vector space ${\displaystyle X}$: Pick a local base ${\displaystyle B}$ for ${\displaystyle X}$ about 0; then (${\displaystyle x_{k}}$) is a Cauchy sequence if for each member ${\displaystyle V\in B,}$ there is some number ${\displaystyle N}$ such that whenever ${\displaystyle n,m>N,x_{n}-x_{m}}$ is an element of ${\displaystyle V.}$ If the topology of ${\displaystyle X}$ is compatible with a translation-invariant metric ${\displaystyle d,}$ the two definitions agree.

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 ${\displaystyle (x_{k})}$ in a topological group ${\displaystyle G}$ is a Cauchy sequence if for every open neighbourhood ${\displaystyle U}$ of the identity in ${\displaystyle G}$ there exists some number ${\displaystyle N}$ such that whenever ${\displaystyle m,n>N}$ it follows that ${\displaystyle x_{n}x_{m}^{-1}\in U.}$ As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in ${\displaystyle G.}$

As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in ${\displaystyle G}$ that ${\displaystyle (x_{k})}$ and ${\displaystyle (y_{k})}$ are equivalent if for every open neighbourhood ${\displaystyle U}$ of the identity in ${\displaystyle G}$ there exists some number ${\displaystyle N}$ such that whenever ${\displaystyle m,n>N}$ it follows that ${\displaystyle x_{n}y_{m}^{-1}\in U.}$ This relation is an equivalence relation: It is reflexive since the sequences are Cauchy sequences. It is symmetric since ${\displaystyle y_{n}x_{m}^{-1}=(x_{m}y_{n}^{-1})^{-1}\in U^{-1}}$ which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since ${\displaystyle x_{n}z_{l}^{-1}=x_{n}y_{m}^{-1}y_{m}z_{l}^{-1}\in U'U''}$ where ${\displaystyle U'}$ and ${\displaystyle U''}$ are open neighbourhoods of the identity such that ${\displaystyle U'U''\subseteq U}$; such pairs exist by the continuity of the group operation.

There is also a concept of Cauchy sequence in a group ${\displaystyle G}$: Let ${\displaystyle H=(H_{r})}$ be a decreasing sequence of normal subgroups of ${\displaystyle G}$ of finite index. Then a sequence ${\displaystyle (x_{n})}$ in ${\displaystyle G}$ is said to be Cauchy (with respect to ${\displaystyle H}$) if and only if for any ${\displaystyle r}$ there is ${\displaystyle N}$ such that for all ${\displaystyle m,n>N,x_{n}x_{m}^{-1}\in H_{r}.}$

Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on ${\displaystyle G,}$ namely that for which ${\displaystyle H}$ is a local base.

The set ${\displaystyle C}$ of such Cauchy sequences forms a group (for the componentwise product), and the set ${\displaystyle C_{0}}$ of null sequences (sequences such that ${\displaystyle \forall r,\exists N,\forall n>N,x_{n}\in H_{r}}$) is a normal subgroup of ${\displaystyle C.}$ The factor group ${\displaystyle C/C_{0}}$ is called the completion of ${\displaystyle G}$ with respect to ${\displaystyle H.}$

One can then show that this completion is isomorphic to the inverse limit of the sequence ${\displaystyle (G/H_{r}).}$

An example of this construction familiar in number theory and algebraic geometry is the construction of the ${\displaystyle p}$-adic completion of the integers with respect to a prime ${\displaystyle p.}$ In this case, ${\displaystyle G}$ is the integers under addition, and ${\displaystyle H_{r}}$ is the additive subgroup consisting of integer multiples of ${\displaystyle p_{r}.}$

If ${\displaystyle H}$ is a cofinal sequence (that is, any normal subgroup of finite index contains some ${\displaystyle H_{r}}$), then this completion is canonical in the sense that it is isomorphic to the inverse limit of ${\displaystyle (G/H)_{H},}$ where ${\displaystyle H}$ varies over all normal subgroups of finite index. For further details, see Ch. I.10 in Lang's "Algebra".

A real sequence ${\displaystyle \langle u_{n}:n\in \mathbb {N} \rangle }$ 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 ${\displaystyle u_{H}}$ and ${\displaystyle u_{K}}$ are infinitely close, or adequal, that is,

${\displaystyle \mathrm {st} (u_{H}-u_{K})=0}$

where "st" is the standard part function.

Krause (2020) introduced a notion of Cauchy completion of a category. Applied to ${\displaystyle \mathbb {Q} }$ (the category whose objects are rational numbers, and there is a morphism from x to y if and only if ${\displaystyle x\leq y}$), this Cauchy completion yields ${\displaystyle \mathbb {R} \cup \left\{\infty \right\}}$ (again interpreted as a category using its natural ordering).