On Bertrand's Postulate

Question

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?

Answer 1

By the PNT, we expect

$$\frac{2n}{\ln(2n)}-\frac{n}{\ln(n)}$$

$$=\frac{2n\ln(n)-n\ln(2n)}{\ln(2n)\ln(n)}$$

$$\sim\frac{n}{\ln(n)}$$

$$\to\infty,\;\; \text{as}\;\; n\to\infty$$

primes in this region.