Proof that $(n!)^2/(2n)!$ converges to $0$. I take following steps:
- $(n!)^2/(2n)(2n-1)\cdots(n!) = (n!)/(2n)(2n-1)\cdots(n-1)$. I assume (do I need to prove?) that $n!$ divides $(2n)(2n-1)\cdots(n-1)$.
- So I have at the end $1/K$ ($K$ is the remainder after division of the denominator by $n!$).
- $1/K$ as increases with increasing $n$ converges to $0$, is a null sequence.
Thanks for any advice.