I am considering the following question:
Suppose $X_1,\cdots,X_n$ are i.i.d. samples from unifrom distribution $U[0,1]$. Let $\hat{F}_n(x)$ be the empirical c.d.f., i.e., $\hat{F}_n(x) = \frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq x\}}$, and $\hat{m}_n = \inf\{x:\hat{F}_n(x)\geq \frac{1}{2}\}$ the estimation of the median. Show that $|\hat{m}_n-m|\overset{a.s.}{\rightarrow}0$, where $m = 1/2$ is the true median.
I searched for the question and I got several general solutions proving that convergence of c.d.f. implies the convergence of the quantiles. I just want to know whether there is any simple solutions for this special case.
What I have done: by drawing the plots of $\hat{F}_n(x)$ and $F(x)=x$, I think we should have $$ |\hat{m}_n-m|\leq \|\hat{F}_n-F\|_{\infty}. $$ If this is true, then I can prove the convergence by the Glivenko-Cantelli theorem $\|\hat{F}_n-F\|_{\infty}\rightarrow 0$. But I got suck on this inequality. Could anyone please give me some ideas on this inequality? Thanks a lot!