Editing Cauchy sequence (section)

===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".