Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a, with limits $\mu_\infty(x_i)$ as defined in 12b. Show that the application $$\begin{aligned} \psi : \mathbb{N}^* &\longrightarrow \mathbb{N}^* \\ k &\longmapsto \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{aligned}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a, with limits $\mu_\infty(x_i)$ as defined in 12b. Show that the application
$$\begin{aligned} \psi : \mathbb{N}^* &\longrightarrow \mathbb{N}^* \\ k &\longmapsto \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{aligned}$$
is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.