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Matematicheskie Zametki, 1973, Volume 14, Issue 6, Pages 809–819 (Mi mzm7299)  

$(i)$-Convergence and its application to a sequence of functions

V. I. Shirokov

Arzamas State Pedagogical Institute
Abstract: Let $(x_\alpha)_{\alpha\in A}$ , where $A$ is a directed set containing cofinal chains — a generalized sequence in a complete chain. It is established that every such sequence contains a monotonic cofinal sub-sequence. For a monotonically increasing (decreasing) bounded sequence $(x_\alpha)_{\alpha\in A}$, by definition, we put $(i)-\lim\limits_{\alpha\in A}x_\alpha=\sup\limits_{\alpha\in A}(x_\alpha)\cdot((i)-\lim\limits_{\alpha\in A}x_\alpha=\inf\limits_{\alpha\in A}(x_\alpha))$. For an arbitrary sequence $(x_\alpha)_\alpha\in A(i)$ the $(i)$-limit is defined as the common $(i)$-limit of its monotonic cofinal sub-sequences. The properties of $(i)$-convergence and some of its applications to generalized sequences of mappings are discussed.
English version:
Mathematical Notes, 1973, Volume 14, Issue 6, Pages 1023–1028
DOI: https://doi.org/10.1007/BF01099585
Bibliographic databases:
UDC: 517.5
Language: Russian
Citation: V. I. Shirokov, “$(i)$-Convergence and its application to a sequence of functions”, Mat. Zametki, 14:6 (1973), 809–819; Math. Notes, 14:6 (1973), 1023–1028
Citation in format AMSBIB
\Bibitem{Shi73}
\by V.~I.~Shirokov
\paper $(i)$-Convergence and its application to a~sequence of functions
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 6
\pages 809--819
\mathnet{http://mi.mathnet.ru/mzm7299}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=418013}
\zmath{https://zbmath.org/?q=an:0285.54024}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 6
\pages 1023--1028
\crossref{https://doi.org/10.1007/BF01099585}
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