Without using the fact that the empirical cdf $F_{n}(x)$ converges to $F(x)$ under higher modes of convergence, I want to show that the empirical cdf converges to F(x) in distribution. In other words, I want to show that
$(1/n)\sum_{k=1}^{n}1_{\{X_{k} \leq x\}} \rightarrow F(x)$ in distribution as $n\rightarrow \infty$. So I want to show that the cdf of $F_{n}(x)$, which I will denote $G_{n}(x)$, converges to $\delta_{F(x)}$, which denotes the cdf of a random variable with a point mass at F(x).
Any hints would be appreciated.