\[n^{\pi(2n)-\pi(n)}< \prod_{n<p\leq 2n\atop p\ \mbox{\footnotesize is prime}}p \leq{2n\choose n}< (1+1)^{2n}= 4^{n}.\]

Peter wrote: Show that $ n^{\pi(2n) - \pi(n)} < 4^{n}$ for all positive integer $ n$. Sorry, what is $ \pi(n)$ ? .

The number of primes $ \leq n$.