Factorial as Product of Consecutive Factorials/Lemma 2

Theorem
Let $n \in \N$.

Then $\paren {2 n - 2}! \, \paren {2 n - 1}! > \paren {3 n - 1}!$ for all $n \ge 7$.

Proof
We prove the result by induction on $n$.

Basis for the Induction
The case $n = 7$ is verified by direct calculation:


 * $12! \times 13! > 20!$

This is the basis for the induction.

Induction Hypothesis
We suppose for some $k \ge 7$, we have:


 * $\paren {2 k - 2}! \, \paren {2 k - 1}! > \paren {3 k - 1}!$

This is our induction hypothesis.

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.