Factorial as Product of Consecutive Factorials/Lemma 1

Theorem
Let $n \in \N$.

Then $\paren {2 n - 1}! \, \paren {2 n}! > \paren {3 n - 1}!$ for all $n > 1$.

Proof
Let $n, k \in \N_{> 0}$.

Suppose $n > 1$ and $n > k$.

We show that $\paren {k + 1} \paren {2 n - k} > 2 n + k$.

For $k = 1$:


 * $2 \paren {2 n - 1} = 4 n - 2 \ge 2 n + 2 > 2 n + 1$

For $k > 1$:

Therefore we have: