The generalization and proof of Bertrand's postulate

Abstract

The purpose of this paper is to show that for 0<r<1 one can determine explicitly an x0 such that ∀x≥x0, ∃ at least one prime between rx and x. This is a generalization of Bertrand's Postulate. Furthermore, the same procedures are used to show that if one can find upper and lower bounds for θ(x) whose difference is kxρ then ∃ a prime between x and x−Kxρ, where k, K>0 are constants, 0<ρ<1 and θ(x)=∑p≤xlnp, where p runs over the primes.