Showing p-th sample quantile is asymptotically normal

Question

I'm working through van der Vaart to improve my knowledge on asymptotic statistics, and I'm attempting the following problem.

Let $F_n^{-1}(p)$ be the p-th sample quantile of a sample from a cumulative distribution $F$ on $\mathbb{R}$ that is differentiable with positive derivative at the population p-th-quantile.

Define $F^{-1}(p) = \inf\{x: F(x) \geq p\}$.

Show that $\sqrt(n)(F_n^{-1}(p) - F^{-1}(p))$ is asymptotically normal with mean 0 and variance $\frac{p(1-p)}{f(F^{-1}(p))^2}$

My attempt

To me, this kind of screamed out using the Delta Method (though it's a problem in the M-statistics section of the book). I considered the empirical CDF $\hat{p} = \sum_{i=1}^{n} I\{x_i \leq x\} / n$. By CLT, we get:

$\sqrt{n}(\hat{p} - p) \overset{d}{\rightarrow} N(0, p(1-p))$.

Using the delta method with $F^{-1}(p)$ seems like it'd give the result I need, but I don't see how $F^{-1}(\hat{p})$ is equal to the p-th sample quantile. Am I misunderstanding something here? Is the use of the empirical CDF as the estimator flawed? I feel like I'm generally on the right track, but I can't make the last leap I need to. Any help would be appreciated.