Let $R$ be a discrete valuation ring that is complete with respect to its maximal ideal $M$,and let $v$ be the valuation on $R$. And $p$ be a prime. And we assume $0<v(p)<∞$.
Why equality $v(n!)=\sum_{i=1}^{\infty} [\dfrac{n}{p^i}]v(p)$ holds?
I know this is true if $v$ is p-adic valuation, this case is elementary.
Thank you in advance.