LIMITS WITHOUT EPSILONS

Abstract

The concept of convergence of real sequences is completely characterized by six properties, each essentially a theorem from the theory of limits of sequences of real numbers. Consequently, the foundation of the calculus can be constructed without the use of the “$ϵ,\delta$” definitions of Cauchy. The six properties are versions of: (1) the scalar multiple of a convergent sequence is convergent; (2) the corresponding sums of equivalent convergent sequences are equivalent; (3) the Squeezing Theorem; (4) all subsequences of a convergent sequence converge and are equivalent; (5) the sequence $1,-1,1,-1,\cdots$ does not converge; and (6) every divergent and bounded sequence has two non-equivalent, convergent subsequences. Five of the six properties are shown to be necessary for the characterization of convergence. The question of the independence of the Squeezing Theorem from the other five properties remains unresolved.