$k$ is nonnegative integer. I want to show that$$ \lim _{n \to \infty} {\frac{n!}{n^k(n-k)!}}=1$$
My try :
$$ \frac{n!}{n^k(n-k)!} = \frac{n}{n} \frac{n-1}{n} \cdots \frac{n-k+1}{n}$$
I wanted use multiplicative rule
If $a_n \to a$ and $b_n \to b$ , then $a_n b_n \to ab$.
But It is impossible because $$ \frac{n!}{n^k(n-k)!} = \frac{n}{n} \frac{n-1}{n} \cdots \frac{n-k+1}{n}$$ It is infinite product.
I want you to help me.