One of my homework problems asks us to show that a sequence of sets of p-quantiles converges to a given set of p-quantiles. I can start the question but don't know how to continue. Here's the problem statement:
Consider a random variable $D$ with cumulative distribution function $F$. Let $D_1$, ..., $D_n$ be an i.i.d. random sample of $D$ with empirical distribution function $H_n(x) = \frac{1}{n} \sum_{i=1}^{n} I${$ D_i \leq x$}, where $I$ represents the indicator function. Consider any $p \in (0,1)$, and let $F^{-1}(\${$p$}$) = [a,b]$, and $H_n^{-1}(${$p$}$) = [a_n,b_n]$. (These intervals represent the p-quantiles of the c.d.f. $F$ and $H_n$, respectively.)
For a set $A \subset \mathbb{R}$ and $x \in \mathbb{R}$, let $d(x, A) := inf${$|x - y| : y \in A$} denote the distance between $x$ and $A$. For two sets $A, B \subset \mathbb{R}$, let $e(A,B) := sup${$d(x,A) : x \in B$} denote the excess of $B$ over $A$.
The question is: "Use the Strong Law of Large Numbers" to show that $P\[e([a_n,b_n],[a,b]) \rightarrow 0$ as $n \rightarrow \infty] = 1$, that is, $[a_n, b_n]$ moves close to $[a,b]$ as $n \rightarrow \infty$."
Beginning of answer:
I first showed that for all $x$, the expectation of $H_n(x)$ was $F(x)$ itself. This stems from the fact that $P\[I${$ D_i \leq x$}$\] = P\[D_i \leq x\] = F(x)$. I then used the Strong Law of Large Numbers to deduce that the sequence of functions {$H_n$}$_{n \in \mathbb{N}}$ converged pointwise to $F$ with probability 1.
Then, I don't really know how to show that the sequence of the sets of p-quantiles {$H_n^{-1}(p)$}$_{n \in \mathbb{N}}$ converges to $F^{-1}(p)$. Using $\varepsilon$, $\delta$ notations, I essentially want to show that $\forall \varepsilon > 0$ there is an $N$ positive such that $e([a_n,b_n],[a,b]) < \varepsilon$ $\forall n \geq N$ with probability 1. This is equivalent to showing that $\forall \varepsilon > 0$ there is an $N$ positive such that $d(x,[a,b]) < \varepsilon$ $\forall n \geq N$ $\forall x \in [a_n,b_n]$ with probability 1. That's where I get stuck. I don't really know how to proceed from here.
Any help is greatly appreciated!