Main Index Mathematical Analysis Infinite series and products Sequences
  Subject Index
comment on the page

Cauchy-Bolzano convergence criterion for sequences

A classical result giving a necessary and sufficient criterion for the convergence of number sequences, which is usually given in one of the following ways:

I. A necessary and sufficient condition for the convergence of the sequence

(1)

is that, for every , there exists a positive integer such that, for every , we have .

II. A necessary and sufficient condition for the convergence of the sequence (1) is that, for every , there exists a positive integer such that, for every and every positive integer we have .

This result of fundamental importance for the theory of convergence was first published by [1]  , p.259.   It was formulated already by Bolzano in 1817,  but his result remained  unknown until Otto Stolz  [2] , [3]  rediscovered many of his articles (cf. also ).

Goursat  [4]  ,p. 8,  gives an equivalent refomulation:

III. A necessary and sufficient condition for the convergence of the sequence (1) is that, for every , there exists a positive integer such that, for every positive integer we have .

Sequences (1)  possessing the property of statement I that, for every , there exists a positive integer such that, for every , we have are called Cauchy sequences .

A natural question connected with statement II asks whether the condition that runs over the set of positive integers can be replaced by a condition that   runs over a proper subset of positive integers . N.Neculce and P.Obreanu  [5]  investigate this question and given some conditions which must fulfill in order to obtain a convergence or divergence criterion when replaces in II. An interesting result proved in  [5]  says that we can take for any set of positive integers which is an asymptotic basis of finite order. One very well known example of such set is the set of primes (a result connected with attempts to solve the Goldbach conjecture).   

References

[1]	Cauchy, A. L. (1821). Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique . Paris: Chez Debure frères.

Cite this web-page as: