Chinese Quarterly Journal of Mathematics ›› 1998, Vol. 13 ›› Issue (3): 74-80.

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An Equivalent Form of the Dedekind Axiomand Its Application(Ⅰ)——Also on theunity ofthe ContinuousInduction,theMathem aticalInduction and the TransfiniteInduction

  

  1.  Department of Mathematics, Guangzhou Teacher's College, Guangzhou, 510400
  • Received:1997-11-20 Online:1998-09-30 Published:2024-10-25
  • Supported by:
    Project supported by the following:① the1352Project Fund of ninth five-year plan made by Higher Education Office of
     Guangdong Province.② the Subject-supporting Fund for Colleges and University of Guangdong Province.③ the Important
     Subject Fund for Colleges and Universities of Guangzhou.The project is included in National Educational Project of1998
    2000.

Abstract: In this paper,a converse and negated form of the continuous induction is indicated. We use it to prove six basic theorems of completeness of real numbers system,the existence theorem of limes superiores and limes inferiores of seqence and the properties of continuous function in closed interval by mode of unification. 

Key words: continuous induction, the Dedekind Axiom, completeness

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