Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge in distribution to a random variable \(X\) where \(X_{n}\) has distribution function \(F_{n}\) for all \(n \in \mathbb{N}\) and \(X\) has distribution function \(F\), which may or may not be a valid distribution function. Prove that if the sequence \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is bounded in probability then

\[\lim _{x ightarrow-\infty} F(x)=0 .\]