Let \(K\) be any non-decreasing right-continuous function such that

\[\begin{gathered}\lim _{t ightarrow \infty} K(t)=1 \\\lim _{t ightarrow-\infty} K(t)=0\end{gathered}\]

and

\[\int_{-\infty}^{\infty} t d K(t)=0\]

Define the kernel estimator of the distribution function \(F\) to be

\[\tilde{F}_{n}(x)=n^{-1} \sum_{k=1}^{n} K\left(X_{i}-xight)\]

Prove that \(\tilde{F}_{n}\) is a valid distribution function.