Proofs from the Book - need quick explanation I've been recently reading this amazing book, namely the chapter on Bertrand's postulate - that for every $n\geq1$ there is a prime $p$ such that $n<p\leq2n$.
As an intermediate result, they prove that $\prod_{p\leq x}p \le 4^{x-1}$ for any real $x\geq2$, where the product is taken over all primes $p\leq x$ . While proving that, they rely on the inequality
$$
\prod_{m+1<p\le2m+1}p\leq\binom{2m+1}{m},
$$
where $m$ is some integer, $p$'s are primes.
They explain it by observing that all primes we are interested in are contained in $(2m+1)!$, but not in $m!(m+1)!$. The last part is what I don't understand.
I can understand how this principle can be applied to the bound $(2m+1)!/(m+1)! = (m + 2)\ldots(2m+1)$, but why can we safely divide this by $m!$?
Thank you!

The answer to the last question is that ${a\choose b}=\frac{a!}{b!(a-b)!}$ is known to be an integer, for all integers $a, b$ satisfying $a\ge b\ge 0$.  The reason is that it is counting the number of ways to choose $b$ out of $a$ objects.  This is necessarily an integer.
