Suppose $X_1,\cdots,X_n$ are i.i.d. continuous random variables from distribution with cdf $F(x)$.Let $F_n(x)$ be a random variable defined by $$F_n(x)=\frac{1}n\sum_{i=1}^nI\{X_i\le x\}.$$ Define the $p$th quantile for cdf $F(x)$ as $$\xi_p\equiv F^{-1}(p)=\inf\{x:F(x)\ge p\},$$ and the $p$th quantile for empirical cdf $F_n(x)$ as $$\hat\xi_p\equiv F_n^{-1}(p)=\inf\{x:F_n(x)\ge p\}.$$ Show that $$\Big|F^{-1}\big(F_n(\xi_p)\big)-\hat\xi_p\Big|\overset{a.s}\to0$$
How can we prove the result? By LLN, we have $F_n(x)\overset{a.s}\to F(x)$. Can we write the equation as $$\Big|F^{-1}\big(F_n(\xi_p)\big)-\hat\xi_p\Big|=\Big|F^{-1}\big(F_n(\xi_p)\big)-F_n^{-1}\big(F(\xi_p)\big)\Big|,$$ and get the result directly?