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Cauchy SequenceIn mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence for which all remaining terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence. Cauchy sequence in a metric space Formal definition Formally, a Cauchy sequence is a sequence -
in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > N, the distance -
is less than r. 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 M. Nonetheless, this may not be the case. Completeness A metric space X in which every Cauchy sequence has a limit (in X) is called complete. Example: real numbers The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers. Counter-example: rational numbers The rational numbers themselves are not complete, for example: - The sequence defined by x1 = 1, xn+1 = (xn + 2/xn)/2 consists of rational numbers (1, 3/2, 17/12,...) which is clear from its definition, and converges to the irrational square root of two.
- The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are irrational for any rational value of x≠0, but are defined as limit of a rational sequence which is their power series.
Other properties Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then is a Cauchy sequence in N. If and are two Cauchy sequences in the rational, real or complex numbers, then the sum and the product are also Cauchy sequences. Cauchy sequences in topological vector spaces There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0; then (xk) is a Cauchy sequence if for all members V of B, there is some number N such that whenever n,m > N, xn - xm is an element of V. If the topology of X is compatible with a translation-invariant metric d, the two definitions agree. Cauchy sequences in groups There is also a concept of Cauchy sequence in a group G: Let H=(Hr) be a decreasing sequence of normal subgroups of G of finite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H) iff for any r there is N such that ∀m,n > N, xn xm-1 ∈ Hr. The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences (s.th. ∀r, ∃N, ∀n > N, xn∈Hr) is a normal subgroup of C. The factor group C/C0 is called the completion of G w.r.t. H. One can then show that this completion is isomorphic to the inverse limit of the sequence (G/Hr). If H is a cofinal sequence (i.e. any normal subgroup of finite index contains some Hr), 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". References
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