Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a, and let $\mu_\infty(x_i)$ be the limits defined in 12b. Show that the map $$\begin{array}{rcl} \psi : \mathbb{N}^* & \longrightarrow & \mathbb{N}^* \\ k & \longmapsto & \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{array}$$ 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 a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a, and let $\mu_\infty(x_i)$ be the limits defined in 12b.
Show that the map
$$\begin{array}{rcl} \psi : \mathbb{N}^* & \longrightarrow & \mathbb{N}^* \\ k & \longmapsto & \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{array}$$
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 $\mu_\infty(x_i)$.