К ОПРЕДЕЛЕНИЮ НЕКОТОРЫХ ПОНЯТИЙ ТЕОРИИ ЧИСЛОВЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ В КЛАССИЧЕСКОМ МАТЕМАТИЧЕСКОМ АНАЛИЗЕ

Abstract

The concept of the limit of a numerical sequence is a long-established concept in the course of mathematical analysis. It is a defined concept and its definition took shape in the form of an "ε – m" definition or definition according to Cauchy. The article shows that the classical definition can be further improved in the sense of a more logical reduction to previously introduced concepts in the most natural form. The definition proposed by the author is called the τ-definition. It is shown that the Cauchy definition and τ-definition are equivalent. By comparing them, certain advantages of the latter were revealed. Some properties of the limit and their new proofs are also considered. The article also gives a new proof of the uniqueness of the limit as well as the boundedness of a convergent numerical sequence and shows the logical vulnerability of the traditional proof. A new definition of a fundamental sequence and a proof of the equivalence of the convergence and fundamental nature of a numerical sequence are proposed. It is shown that for sign-alternating numerical sequences the concept of conditional convergence does not make sense, although this concept exists for sign-alternating numerical sequences. Comparison criteria for non-negative sequences of sums are introduced, preceding the same criteria for non-negative numerical series.