How can one show that $\prod_{nI am trying to rove that $\prod_{n<p\leq2n}p \leq C(2n,n) \leq 2^{2n}$, 
where  $C(2n,n)= \frac{2n!}{n! n!}$ and $p$ is prime.
I can prove the second part by induction, but first part induction doesn't work. 
I can see that $\prod_{n<p\leq 2n}p \leq \frac{2n!}{n!}$ but there is an extra $n!$ in the denominator. 
Any suggestion?

Note that $p \mid (2n)!$ and since $p$ is prime and greater than $n$ it does not divide $n!$. 
From there you can derive that each $p$, and thus also the product of all, in fact divides the binomial coefficient. 
