Bertrand's postulate in another point of view I was just wondering why can't we use prime number's theorem to prove Bertrand's postulate.We know that if we show that for all natural numbers $n>2, \pi(2n)-\pi(n)>0$ we are done. Why can't it be proven by just showing (By using the prime number's theorem) that for every natural numbers $n>2, \frac{2n}{ln(2n)}-\frac{n}{ln(n)}>0$?

The following long-standing conjecture, due to Legendre, has a flavour similar to Bertrand's Postulate. For every $n\gt 1$, there is always a prime between $n^2$ and $(n+1)^2$. 
By using the Prime Number Theorem, we can see that "usually," for large $n$, there are many primes between $n^2$ and $(n+1)^2$. However, the PNT is consistent with substantial irregularities of distribution, and is not strong enough to settle Legendre's conjecture. 
