ABSTRACT
The existence of the limit of the numerical sequences can be determined by the Cauchy criterion. The majority of the mathematical spaces are non-complete. The concept of completeness is connected with the natural desire to have unconditional fulfillment of some property, in this case, the convergence of fundamental sequences. The additive equation on the set of natural numbers is “incomplete” in a certain sense, and “complete” on the set of integers. The spaces of rational or positive numbers, continuous functions with the integral metric and Riemann integrable functions are not complete. The most important classes of the complete metric spaces are the Banach spaces and the Hilbert spaces. The Cauchy criterion is a very effective method of proving the convergence of the sequences. This is applicable not only for numerical sequences, but for classes of functional sequences too.
