![]() ![]() ![]() Such sequences are called Cauchy sequences. More stringent means you should start out with the definition of what you want to show (or some equivalent statement) and then set up the quantities in your reasoning accordingly. Cauchy sequences are named after the French mathematician Augustin Louis Cauchy, 1789-1857. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers. begingroup RickyNelson Look at the answer from Doug M. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers has the real limit x. The notions above are not as unfamiliar as they might at first appear. The concepts of real analysis underpin calculus and its application to it. Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Using the definition of a Cauchy sequence, prove that (1/n2) is a Cauchy sequence. The utility of Cauchy sequences lies in the fact that in a complete metric space, the criterion for convergence depends only on the terms of the sequence itself. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. ![]() In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Freebase Rate this definition: 0.0 / 0 votes ![]()
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