Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n

Theorem
Let $\dbinom {2 n} n$ denote a binomial coefficient.

Then for all prime numbers $p$ such that $n < p < 2 n$:


 * $p \divides \dbinom {2 n} n$

where $\divides$ denotes divisibility.

Proof
By definition of binomial coefficient:

Let $\mathbb P$ denote the set of prime numbers.

Let $p \in \mathbb P$ such that $n < p < 2 n$.

By definition of factorial:
 * $p \divides \paren {2 n}!$

That is:
 * $p \divides \dbinom {2 n} n \paren {n!}^2$

From Prime iff Coprime to all Smaller Positive Integers:
 * $p \nmid n!$

where $\nmid$ denotes non-divisibility.

and so:
 * $p \nmid \paren {n!}^2$

But from Euclid's Lemma for Prime Divisors, $p$ is a divisor of either $\dbinom {2 n} n$ or $\paren {n!}^2$.

Hence it must be the case that:
 * $p \divides \dbinom {2 n} n$