On Flexible Sequences


In the setting of nonstandard analysis, we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical O(⋅) and o(⋅) notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions, and multiplications. We introduce two forms of convergence and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications.

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The first author acknowledges the support of the Centro de Matemática, Aplicações Fundamentais e Investigação Operacional / Fundação da Faculdade de Ciências da Universidade de Lisboa via the grant UID/MAT/04561/2013 and a postdoc-grant from Erasmus Mundus Mobility with Asia–East 14.

The second author acknowledges a PhD-grant of Erasmus Mundus Mobility with Asia–East 14.

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Correspondence to Bruno Dinis.

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Dinis, B., Van Tran, N. & Berg, I.v.d. On Flexible Sequences. Acta Math Vietnam 44, 833–874 (2019). https://doi.org/10.1007/s40306-018-00303-4

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  • External numbers
  • Flexible sequences
  • Convergence
  • Nonstandard analysis

Mathematics Subject Classification (2010)

  • 03H05
  • 40A05