A Classical Olivier's Theorem and Statistical ConvergenceTibor Šalát Vladimír Toma Abstract L. Olivier proved in 1827 the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. We prove that this result remains true if we omit the monotonicity of the terms of the series when the limit operation is replaced by the statistical limit, or some generalizations of this concept.Résumé. L. Olivier démontrait en 1827 un résultat classique sur la vitesse de convergence vers zéro d'une série convergente à termes positifs décroissants. Nous démontrons que ce résultat reste valable si nous omettons la monotonie des termes de la série, en remplaçant l'opération limite par la limite statistique ou encore par des générali-sations de ce concept.
Abstract. In this paper we introduce the notion of I K -Cauchy function, where I and K are ideals on the same set. The I K -Cauchy functions are a generalization of I * -Cauchy sequences and I * -Cauchy nets. We show how this notion can be used to characterize complete uniform spaces and we study how I K -Cauchy functions and I-Cauchy functions are related. We also define and study I K -divergence of functions.
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