Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 833–874 | Cite as

On Flexible Sequences

  • Bruno DinisEmail author
  • Nam Van Tran
  • Imme van den Berg


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.


External numbers Flexible sequences Convergence Nonstandard analysis 

Mathematics Subject Classification (2010)

03H05 40A05 



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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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Bruno Dinis
    • 1
    Email author
  • Nam Van Tran
    • 2
  • Imme van den Berg
    • 3
  1. 1.Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisboaPortugal
  2. 2.Faculty of Applied SciencesHo Chi Minh City University of Technology and EducationHo Chi Minh CityVietnam
  3. 3.Departamento de MatemáticaUniversidade de ÉvoraÉvoraPortugal

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