The Cauchy criterion and real limiting processes

Abstract

The paper develops the principal theorems on limits of convergent sequences starting from the Cauchy criterion. The traditional definition causes problems to students as it presupposes the existence of a limit before checking that a sequence converges. The Cauchy criterion, in addition to overcoming this problem, also leads to more precise results. The convergence of sequences generated on a computer is also examined and practical criteria for terminating iterations are compared and analysed.