Suppose we have two random variable $X$ and $Y$. The conditional CDF of $Y|X$ is defined as $F(y|X)=P(Y\le y|X)$. Let $X_1,X_2,\cdots, X_n$ be random sample of $X$. Show that $$\sup_y\left|\frac1n\sum_{i=1}^n F(y|X_i)-E(F(y|X))\right|\overset{p}\to0.$$
By LLN, we can obtain that $$\frac1n\sum_{i=1}^n F(y|X_i)-E(F(y|X))\overset{a.s.}\to0$$ My question is that how can we obtain the uniform convergence?