In this theorem, Jun Shao proved some results about the asymptotic distribution for $\sqrt{n}(\hat{\theta}_p - \theta_p)$, where $\theta_p = F^{-1}(p)$ and $\hat{\theta}_p = F^{-1}_n(p)$ denote the $p$th quantile of F and the $p$th sample quantile, respectively, and $F$ is a c.d.f. from which the i.i.d. samples $X_1, \cdots, X_n$ are from.
The theorem states that
If $F$ is continuous at $\theta_p$ and there exists $F'(\theta_p+) > 0$, then $P(\sqrt{n}(\hat{\theta}_p - \theta_p) \leq t) \rightarrow \Phi(t/\sigma_F^+)$, where $\sigma_F^+ = \sqrt{p(1-p)}/F'(\theta_p+)$.
In the proof, Shao first defined $t > 0, p_{nt} = F(\theta_p + t\sigma_F^+ n^{-1/2})$, and $c_{nt} = \sqrt{n}(p_{nt} - p) / \sqrt{p_{nt}(1-p_{nt})}$, then he claimed that under the assumed conditions on $F$, $p_{nt} \rightarrow p$, and $c_{nt} \rightarrow t$.
I can understand why $p_{nt} \rightarrow p$, but my question is, why is it that $c_{nt} \rightarrow t$?
My intuition tells me to use the central limit theorem (given the form of $c_{nt})$, but I am not sure how to apply it in this case.
For reference, here's the screenshot of the complete proof of the theorem:
