Let \(g\) be a continuous and bounded function and let \(\left\{F_{n}ight\}_{n=1}^{\infty}\) be a sequence of distribution functions such that \(F_{n} \leadsto F\) as \(n ightarrow \infty\) where \(F\) is a distribution function. Prove that when \(b\) is a finite continuity point of \(F\), that

\[\lim _{n ightarrow \infty}\left|\int_{b}^{\infty} g(x) d F_{n}(x)-\int_{b}^{\infty} g(x) d F(x)ight|=0\]