Let \(\left\{F_{n}ight\}_{n=1}^{\infty}\) be a sequence of distribution functions that converge in distribution to a distribution function \(F\) as \(n ightarrow \infty\). Prove that

\[\lim _{n ightarrow \infty} \sup _{x \in \mathbb{R}}\left|F_{n}(x)-F(x)ight|=0\]