Convergence of Sequences

Abstract

Throughout this chapter, “sequence” will mean “real-valued sequence with domain $\dot N$ ” (cf. IV.3.7 and VI.5.3), that is, “element of $R^{\dot N}$ ”; and such sequences will be denoted by u , v ,… . The aim is to define and make use of the concept of convergence of such sequences in a manner and to an extent suggested by top echelon secondary and early tertiary work. It is here that one gets down to what most mathematicians regard as the analytical component of “real mathematics” . By conscious choice, explicit links with the formal background will for the most part be permitted to decline to a more conventional level, but the implicit links are as strong as ever. Occasional reversions to formalities will serve as illustrations of what might (perhaps should) be done all the time.