[Notations] Sets of Continuous functions.
$C_K\,$: the set of continuous functions $f$ each vanishing outside a compact set $K(f)$.
$C_0\;\,$: the set of continuous functions $f$ such that $\lim_{|x|\rightarrow\infty}f(x)=0$.
$C_B\,$: the set of bounded continuous functions.
$C\;\;\,$: the set of continuous functions.
It is clearly that $f\in C_K\implies f\in C_0\implies f\in C_B\implies f\in C$.
[Theorem] Let $\{\mu_n\}_{n\geq1}$ and $\mu$ be a sequence of p.m.'s, then $$\mu_n\overset{v}{\longrightarrow}\mu\iff\forall\,f\in C_B,\;\int f\,d\mu_n\rightarrow\int f\,d\mu.$$
$\bullet$ Proof.
$(\Longrightarrow)$ Suppose $\mu_n\overset{v}{\rightarrow}\mu$. Given $\varepsilon>0$ and let $a,\,b\in D$ dense in $\mathbb{R}$ such that $\mu(a,\,b]^c=1-\mu(a,\,b]<\varepsilon$. By the vague convergence, we have $$\exists\,n_0(\varepsilon)\mbox{ such that for }n\geq n_0(\varepsilon),\;\mu_n(a,\,b]^c=1-\mu_n(a,\,b]<\varepsilon.$$Since $f\in C_B\implies\exists\,M>0\mbox{ such that }|f|\leq M<\infty$ and define $$f_\varepsilon=\begin{cases}f,& x\in[a,\,b];\\ 0,&x<a-1\mbox{ or }x>b+1;\\ \mbox{linear increase or decrease},&\mbox{o.w.}.\end{cases}$$Then, we have
(1) $f_\varepsilon\in C_K\implies\int_\mathbb{R}f_\varepsilon\,d\mu_n\rightarrow\int_\mathbb{R}f_\varepsilon\,d\mu.$ [See Detail]
(2) $|f-f_\varepsilon|\leq 2M\implies$ $$\int_\mathbb{R}|f-f_\varepsilon|\,d\mu_n=\int_{(a,\,b]^c}|f-f_\varepsilon|\,d\mu_n\leq2M\mu_n(a,\,b]^c<2M\varepsilon.$$
(3) Similarly, $\int_\mathbb{R}|f-f_\varepsilon|\,d\mu<2M\varepsilon.$
Thus, $$\begin{array}{rl}\left|\int f\,d\mu_n-\int f\,d\mu\right| & =\left|\int f\,d\mu_n-\int f_\varepsilon\,d\mu_n+\int f_\varepsilon\,d\mu_n-\int f_\varepsilon\,d\mu+\int f_\varepsilon\,d\mu-\int f\,d\mu\right| \\ & \leq\int\left|f-f_\varepsilon\right|\,d\mu_n+\left|\int f_\varepsilon\,d\mu_n-\int f_\varepsilon\,d\mu\right|+\int\left|f-f_\varepsilon\right|\,d\mu \\ &<4M\varepsilon,\,\forall\,\varepsilon>0\end{array}$$due to (1), (2) and (3). We have $$\mu_n\overset{v}{\longrightarrow}\mu\implies\forall\,f\in C_B,\;\int f\,d\mu_n\rightarrow\int f\,d\mu.$$
$(\Longleftarrow)$ Define $g=I(a,b]$. We construct two functions in $C_K$, of course, they are in $C_B$ as well and complete the proof by following the $(\Longleftarrow)$ part in Converge in Distribution and Vague Convergence (1): Equivalence for s.p.m.'s.
$(\Longleftarrow)$ Define $g=I(a,b]$. We construct two functions in $C_K$, of course, they are in $C_B$ as well and complete the proof by following the $(\Longleftarrow)$ part in Converge in Distribution and Vague Convergence (1): Equivalence for s.p.m.'s.
$\Box$
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