State and prove Dynkin's $\pi$-system lemma.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\left(A_{n}\right)$ be a sequence of independent events such that $\lim _{n \rightarrow \infty} \mathbb{P}\left(A_{n}\right)=p$. Let $\mathcal{G}=\sigma\left(A_{1}, A_{2}, \ldots\right)$. Prove that

$$\lim _{n \rightarrow \infty} \mathbb{P}\left(G \cap A_{n}\right)=p \mathbb{P}(G)$$

for all $G \in \mathcal{G}$.