This paper proposes a genetic development of the concept of limit of a sequence leading to a definition, through a succession of proofs rather than through a succession of sequences or a succession of epsilons. The major ideas on which it is based are historical and depend on Euclid, Archimedes, Fermat, Wallis and Newton. Proofs of equality by means of inequalities precede the notion of limit. For example, the determination of the volume of a pyramid precedes the definition of limit. The algebraic details given here are anachronistically modern, and the notion of a vice between two inequalities which provides the distinctive perspective of this paper would not have been recognized in this form by any of the named mathematicians since it presumes the existence of negative numbers and their comparability in a modern sense with positive numbers.