Let $X_1, ..., X_n$ be iid continuous random variables with cdf $F$ and define $Y_n(x) \equiv \sum_{i=1}^n 1(X_i \leq x)$
Define the inverse cdf as $F^{-1}(y) = \text{inf}\{x \in \mathbb{R} : F(x) \ge y\}$, and $[t] = $ $t$ rounded to the nearest integer.
Finally denote $p \equiv F(x)$.
I wish to show that $X_{[np]} - F^{-1}(Y_n(x)) \xrightarrow{P} 0$
Intuitively this makes perfect sense, since $Y_n$ is an approximation of $p$ and the smallest value for which the cdf exceeds $p$ is the true population quantile, while $X_{[np]}$ approximates this.
Any ideas on how to prove this?