Let $F,F_1,F_2,\dots$ be distribution functions. If $F_n$ converges weakly 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 $g:\mathbb{R}\rightarrow\mathbb{R}$ .

Remark. The integrals involved are Riemann-Stieltjes integrals.