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 $E$. We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying condition (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$. We fix $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon$ as in 10a. Show that for every subset $A$ of $E$: $$\left|\mu_n(A) - \mu(A)\right| \leqslant \left|\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)\right| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$ and deduce that if $n \geqslant N_\varepsilon$, then $\left|\mu_n(A) - \mu(A)\right| < 4\varepsilon$.
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 $E$. We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying condition (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$. We fix $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon$ as in 10a. Show that for every subset $A$ of $E$:
$$\left|\mu_n(A) - \mu(A)\right| \leqslant \left|\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)\right| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$
and deduce that if $n \geqslant N_\varepsilon$, then $\left|\mu_n(A) - \mu(A)\right| < 4\varepsilon$.