# On Bertrand's Postulate

By Bertrand's postulate, we know that there exists at least one prime number between $$n$$ and $$2n$$ for any $$n > 1$$. In other words, we have $$\pi(2n) - \pi(n) \geq 1,$$ for any $$n > 1$$. The assertion we would like to prove is that the number of primes between $$n$$ and $$2n$$ tends to $$\infty$$ , if $$n \to \infty$$, that is, $$\lim_{n\to\infty} \pi(2n) - \pi(n) = \infty.$$ Do you see an elegant proof?

• If a proof using the prime number theorem qualifies, there should be an easy and elegant proof. – Peter Mar 19 '20 at 11:01
• I've shown $\frac{\pi(2n)}{\pi(n)} \to 2$ using PNT. – bozcan Mar 19 '20 at 11:07

$$\frac{2n}{\ln(2n)}-\frac{n}{\ln(n)}$$
$$=\frac{2n\ln(n)-n\ln(2n)}{\ln(2n)\ln(n)}$$
$$\sim\frac{n}{\ln(n)}$$
$$\to\infty,\;\; \text{as}\;\; n\to\infty$$