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For the old question whether there is always a prime in the interval [kn,(k+1)n] or not, the famous Bertrand's postulate gave an affirmative answer fork=1. It was first proved by P.L. Chebyshev in 1850, and an elegant elementaryproof was given by P. Erdos in 1932. M. El Bachraoui used elementary techniquesto prove the case k=2 in 2006. This paper gives a proof of the case k=3, againwithout using the prime number theorem or any deep analytic result. In additionwe give a lower bound for the number of primes in the interval [3n, 4n], whichshows that as n tends to infinity, the number of primes in the interval [3n,4n] goes to infinity.

On the Primes in the Interval [3n, 4n]