Let \(\left\{F_{n}ight\}_{n=1}^{\infty}\) be a sequence of distribution functions and let \(F\) be a distribution function such that for each bounded and continuous function \(g\),

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

Prove that if \(\varepsilon>0\) and \(t\) is a continuity point of \(F\), then

\[\liminf _{n ightarrow \infty} F_{n}(t) \geq F(t-\varepsilon) .\]