Editing Cauchy sequence (section)

==Generalizations==

===In topological vector spaces===
There is also a concept of Cauchy sequence for a [[topological vector space]] <math>X</math>: Pick a [[local base]] <math>B</math> for <math>X</math> about 0; then (<math>x_k</math>) is a Cauchy sequence if for each member <math>V\in B,</math> there is some number <math>N</math> such that whenever 
<math>n,m > N, x_n - x_m</math> is an element of <math>V.</math> If the topology of <math>X</math> is compatible with a [[translation-invariant metric]] <math>d,</math> 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 <math>(x_k)</math> in a topological group <math>G</math> is a Cauchy sequence if for every open neighbourhood <math>U</math> of the [[Identity element|identity]] in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n x_m^{-1} \in U.</math> As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in <math>G.</math>

As in the [[Complete metric space#Completion|construction of the completion of a metric space]], one can furthermore define the binary relation on Cauchy sequences in <math>G</math> that <math>(x_k)</math> and <math>(y_k)</math> are equivalent if for every open [[Neighbourhood (mathematics)|neighbourhood]] <math>U</math> of the identity in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n y_m^{-1} \in U.</math> This relation is an [[equivalence relation]]: It is reflexive since the sequences are Cauchy sequences. It is symmetric since <math>y_n x_m^{-1} = (x_m y_n^{-1})^{-1} \in U^{-1}</math> which by continuity of the inverse is another open neighbourhood of the identity. It is [[Transitive relation|transitive]] since <math>x_n z_l^{-1} = x_n y_m^{-1} y_m z_l^{-1} \in U' U''</math> where <math>U'</math> and <math>U''</math> are open neighbourhoods of the identity such that <math>U'U'' \subseteq U</math>; such pairs exist by the continuity of the group operation.

===In groups===

There is also a concept of Cauchy sequence in a [[group (mathematics)|group]] <math>G</math>:
Let <math>H=(H_r)</math> be a decreasing sequence of [[normal subgroup]]s of <math>G</math> of finite [[Index of a subgroup|index]].
Then a sequence <math>(x_n)</math> in <math>G</math> is said to be Cauchy (with respect to <math>H</math>) if and only if for any <math>r</math> there is <math>N</math> such that for all <math>m, n > N, x_n x_m^{-1} \in H_r.</math>

Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on <math>G,</math> namely that for which <math>H</math> is a local base.

The set <math>C</math> of such Cauchy sequences forms a group (for the componentwise product), and the set <math>C_0</math> of null sequences (sequences such that <math>\forall r, \exists N, \forall n > N, x_n \in H_r</math>) is a normal subgroup of <math>C.</math> The [[factor group]] <math>C/C_0</math> is called the completion of <math>G</math> with respect to <math>H.</math>

One can then show that this completion is isomorphic to the [[inverse limit]] of the sequence <math>(G/H_r).</math>

An example of this construction familiar in [[number theory]] and [[algebraic geometry]] is the construction of the [[p-adic number|<math>p</math>-adic completion]] of the integers with respect to a [[prime number|prime]] <math>p.</math> In this case, <math>G</math> is the integers under addition, and <math>H_r</math>  is the additive subgroup consisting of integer multiples of <math>p_r.</math> 

If <math>H</math> is a [[Cofinal (mathematics)|cofinal]] sequence (that is, any normal subgroup of finite index contains some <math>H_r</math>), then this completion is [[Canonical form|canonical]] in the sense that it is isomorphic to the inverse limit of <math>(G/H)_H,</math> where <math>H</math> varies over {{em|all}} normal subgroups of finite [[Index of a subgroup|index]]. For further details, see Ch. I.10 in [[Serge Lang|Lang]]'s "Algebra".

===In a hyperreal continuum===
A real sequence <math>\langle u_n : n \in \N \rangle</math> has a natural [[Hyperreal number|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 <math>u_H</math> and <math>u_K</math> are infinitely close, or [[Adequality|adequal]], that is,
:<math>\mathrm{st}(u_H-u_K)= 0</math> 
where "st" is the [[standard part function]].

===Cauchy completion of categories===
{{harvtxt|Krause|2020}} introduced a notion of Cauchy completion of a [[Category (mathematics)|category]]. Applied to <math>\Q</math> (the category whose [[object (category theory)|objects]] are rational numbers, and there is a [[morphism]] from ''x'' to ''y'' if and only if <math>x \leq y</math>), this Cauchy completion yields <math>\R\cup\left\{\infty\right\}</math> (again interpreted as a category using its natural ordering).