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. Analogous definitions can be given for sequences of natural numbers, integers, etc. Often sequences such as these are called real sequences, sequences of real numbers or sequences in R to make it clear that the elements of the sequence are real numbers.
A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive number, a number can be found such that each of the functions differs from by no more. The notions above are not as unfamiliar as they might at first appear. Definition A sequence of real numbers is any function a : N R. In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. 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. 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. In this paper we prove a decomposition theorem for I-convergent sequences and we introduce the notions of I Cauchy sequence and I -Cauchy sequence, and then. 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.
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