For every integer $d\geq 2$, we denote by $\mathcal{P}(d)$ the set of prime numbers dividing $d$. Show the inequality $$d \geq |\mathcal{P}(d)|!$$ Deduce that $$|\mathcal{P}(d)| = \underset{d\rightarrow+\infty}{o}(\log(d)).$$
For every integer $d\geq 2$, we denote by $\mathcal{P}(d)$ the set of prime numbers dividing $d$.
Show the inequality
$$d \geq |\mathcal{P}(d)|!$$
Deduce that
$$|\mathcal{P}(d)| = \underset{d\rightarrow+\infty}{o}(\log(d)).$$