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The Wikipedia page on convergence of measures says:

In the case $S=\mathbb{R}$ with its usual topology, if $F_n, F$ denote the cumulative distribution functions of the measures $P_n$, $P$ respectively, then $P_n$ converges weakly to $P$ if and only if $\lim_{n\rightarrow\infty} F_n(x) = F(x)$ for all points $x\in\mathbb{R}$ at which $F$ is continuous.

I'm interested in the proof of the following:

If $\lim_{n\rightarrow\infty} F_n(x) = F(x)$ for all points $x\in\mathbb{R}$ at which $F$ is continuous, then $$\lim_{n\rightarrow\infty}\int_\mathbb{R}gdF_n=\int_\mathbb{R}gdF$$ for any function $g\in C_0^\infty(\mathbb{R})$

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Integrate by parts; for each $n$ you have $$\int_{-\infty}^\infty g(x)\,dF_n(x) = - \int_{-\infty}^\infty g'(x) F_n(x)\, dx$$ Now I think you can do the rest.

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