I am trying to follow the CLT proof of Wikipedia's article on CLT
I can follow up to and including the part that $Z_{n} \to N(0,1)$ as $n \to \infty$.
I cannot understand the very last sentence of the proof. How do I go from the distribution of $Z_{n}=\frac{X_1+\cdots+X_n - n \mu}{\sqrt{n \sigma^2}}=\frac{(\sum_{i=1}^{n} X_{i})-n\mu }{\sqrt{n \sigma ^{2} }} \sim N(0,1)$ as $n \to \infty$ to the distribution of $\sum_{i=1}^{n} X_{i} \sim N(n\mu, n\sigma^{2}) $ to the distribution of $\bar{X}\sim N(\mu, \frac{\sigma^{2}}{n})$ ?