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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).
[1] | Cauchy, A. L. (1821). Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique . Paris: Chez Debure frères. ![]() |
[2] | Stolz, O. (1880). B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung. Wien. Anz., 91-92. |
[3] | Stolz, O. (1881). R. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung. Math. Ann., 18, 255-279. |
[4] | Goursat, E. (1910). Course d’Analyse Mathématique, vol. 1 (2nd ed.). Paris. |
[5] | Neculce, N., & Obreanu, P. (1961). The 'weakening' of Cauchy's convergence criterion. Amer. Math. Monthly, 68, 880-886. |
Cite this web-page as:
Štefan Porubský: Cauchy-Bolzano convergence criterion for sequences.