In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used:
$$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$$
(In the original denoted $\text{ord}$ $_{p}(\cdot)$ instead of $\nu_p(\cdot)$).
where $\nu_{p}(k)$ is the $p$-adic valuation of $k\in\mathbb{Q}$, i. e. the exponent of the prime $p$ in the prime factorization of $k$. I know some properties of the foor function and that
$$\nu_{p}(a/b)=\nu_{p}(a)-\nu_{p}(b),$$
$$\nu_{p}(a\cdot b)=\nu_{p}(a)\cdot \nu_{p}(b)$$
and
$$\nu_{p}(n!)=\displaystyle\sum_{i\geq 1}\left\lfloor \dfrac{n}{p^{i}}\right\rfloor $$
but I didn't convince myself on the correct argument I should use to prove $(\ast )$.
Question: How can this inequality be proven?