![]() ![]() There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0 then ( x k) is a Cauchy sequence if for all members V of B, there is some number N such that whenever If (x_n) and (y_n) are two Cauchy sequences in the rational, real or complex numbers, then the sum (x_n + y_n) and the product (x_n y_n) are also Cauchy sequences.Ĭauchy sequences in topological vector spaces If f \colon M \rightarrow N is a uniformly continuous map between the metric spaces M and N and ( x n) is a Cauchy sequence in M, then (f(x_n)) is a Cauchy sequence in N. 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 Maclaurin series.Įvery convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded.The sequence defined by x 0 = 1, x n+1 = ( x n + 2/ x n)/2 consists of rational numbers (1, 3/2, 17/12.), which is clear from the definition it converges to the irrational square root of two, see Babylonian method of computing square root. ![]() There are sequences of rationals that converge (in R) to irrational numbers these are Cauchy sequences having no limit in Q. The rational numbers Q are not complete (for the usual distance): The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers. Nonetheless, this may not be the case.Ī metric space X in which every Cauchy sequence has a limit (in X) is called complete. 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. 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 Cauchy sequence in a metric space Formal definitionįormally, a Cauchy sequence is a sequence x_1, x_2, x_3, \ldots
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