ABSTRACT
This chapter discusses the study of metric spaces and normed vector spaces in relation to the property of completeness or its failure. It examines more closely those spaces and subspaces which are complete, establishing along the way several important results valid on spaces which possess this favourable property of completeness. The phenomenon suggests that it would be beneficial to have a method to determine whether a sequence converges by means of a condition that depends only on the terms of the sequence itself, and not on the a priori knowledge of a potential limit. It is likely that have encountered in the study of sequences of real numbers the concept of a Cauchy sequence which is equivalent in that setting to that of a convergent sequence. The chapter establishes two relatively simple but extremely important results which hold true on complete metric spaces. Both of them are of "existential" flavour, namely they guarantee the existence of some object.
