I use the Empirical Cumulative Distribution Function (EDCF) $$ \hat{F}_n (x)=\frac{1}{n} \sum_{i=1}^{n} \mathbb{1}_{\{X_i \leq x\}}. $$ I know that $Y := n \hat{F}_n (x) \sim Bin(n,p)$ with $p := F(x)$. Let $\hat{p}_n := \frac{Y}{n}$, then it should follow by the Central Limit Theorem that $$ \sqrt{n} \frac{\hat{p}_n-p}{\sqrt{p(1-p)}} \overset{d}{\rightarrow} N(0,1). $$
Can somebody show me the steps how to get there? I start of with the CLT $$ \sqrt{n} (Y_n-np) \overset{d}{\rightarrow} N(0,np(1-p)), $$ and after a few steps I get the result above without the root in the denominator on the left side.
If $(Z_n)$ is a sequence of iid random variable with mean $\mu$ and variance $\sigma^2$ then $\overline Z_n:=\frac1{n}\sum_{i=1}^nZ_i$ has mean $\mu$ and variance $\frac1{n^2}\sum_{i=1}^n\mathsf{Var}(Z_i)=\frac{\sigma^2}n$.
Then: $$U_n:=\frac{\overline{Z_n}-\mathbb E\overline{Z_n}}{\sqrt{\mathsf{Var}(\mathbb E\overline{Z}_n)}}=\frac{\sqrt{n}(\overline{Z_n}-\mu)}{\sigma}$$has mean $0$ and variance $1$ and according to the CLT we have $U_n\stackrel{d}{\to}N(0,1)$ where $N(0,1)$ has standard normal distribution.
Apply this on $Z_i=\mathbf1_{X_i\leq x}$.