Primes in intervals

In this paper, we prove: (a) for every integer n > 1 and a fixed integer k ≤ n, there exists a prime number p such that kn ≤ p ≤ (k + 1)n, and (b) conjectures of Legendre, Oppermann, Andrica, Brocard, and Improved version of Legendre conjecture as a particular case of (a).