Another inequality question

Question

Can somebody in elementary way show that $(n!)^{2\over n+1}>n-1$ for only finitely many $n\in\mathbb N$? I need to prove this to be able to prove something else.

Answer 1

I don't think it's true. It's known that $\log(n!)$ is very roughly $n\log n$, so the log of the left side is very roughly $2\log n$, while the log of the right side is roughly $\log n$, which is smaller. This suggests the inequality holds for all but finitely many $n$.