Limit of p-adic numbers

Question

Let $\alpha\in\mathbb{Z}_p^\times$. I read somewhere that the limit of $\alpha^{n!}$ as $n\rightarrow\infty$ is equal to one. Can someone explain to me why this is? Thanks!

Answer 1

Hint : $a^k\equiv 1\ (\ mod\ p\ )$ , whenever $p-1|k$

Now, use $p-1|n!$ for all $n\ge p-1$.