Helly-Bray theorem

Let $F,F_{1},F_{2},\dots$ be distribution functions. If $F_{n}$ converges weakly (http://planetmath.org/ConvergenceInDistribution) to $F$ , then

\int_{\mathbb{R}}g(x)dF_{n}(x)\xrightarrow[n\rightarrow\infty]{}\int_{\mathbb{% R}}g(x)dF(x)

for each continuous bounded function $g:\mathbb{R}\rightarrow\mathbb{R}$ .