structural:summarization, structural:expansion, stylistic:tone
In groups

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

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

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

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

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