Let $X_n$ iid RVs with the distribution function $F$ which has a density. Let $F_n(x)= \frac{1}{n} \sum_{i=1}^nI(X_i \leq x)$ be the empirical distribution function.
Show $$sup_x \sqrt{n}|F_n(x) -F(x)|=O_p(1)$$.
I know by Glivenko-Cantelli thm that $sup_x |F_n(x) -F(x)| \to_{n \to \infty} 0 \ \text{a.s.}$.
I also know that for each $x\in \mathbb R , \ \sqrt{n}|F_n(x) -F(x)| \to _d N(0,F(x)(1-F(x))$ implying $\sqrt{n}|F_n(x) -F(x)|=O_p(1)$.
I suppose I need to combine those two facts and the fact that $F$ has a density, but I cannot proceed further.
Any hint is appreciated.
Donsker’s Theorem and a functional continuous mapping theorem will do the trick. All you need to show is that the sup map from the functional space to the real line is continuous.