If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the application $$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$ where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that $$\forall x \in \mathbb{N}^*, \quad \mathbf{P}(X = x) = \lim_{n \rightarrow +\infty} \mathbf{P}(\psi_n(X) = x)$$
If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the application
$$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$
where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order. Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that
$$\forall x \in \mathbb{N}^*, \quad \mathbf{P}(X = x) = \lim_{n \rightarrow +\infty} \mathbf{P}(\psi_n(X) = x)$$