Let $p$ be a prime number.
In order to prove a result on $p$-adic interpolation of iterates, I need to show the following:
Lemma. Let $m$ be an integer, one has: $$v_p(m!)\leqslant\frac{m}{p-1}.$$
At that moment I only know that: $$v_p(m!)=\sum_{i=1}^mv_p(i).$$
If possible I would prefer to avoid using Legendre's formula. Thank you in advance for your help!