Prove that (product of primes between n and 2n) > 2^n

I've read that a weak but explicit version of the Prime Number Theorem states that the product of primes between $\displaystyle n$ and $\displaystyle 2n$ (inclusive) is at least $\displaystyle 2^n$ for all $\displaystyle n \geq 1$. I don't think this proof can be elementary (although if someone can prove it below then that would be great!) but is this a standard result? If so, where can I find a proof? I've searched but haven't managed to find anything.