Central Limit Theorem different versions

Question

Which one is correct about the Central Limit Theorem:

1. If $X_{1}, ... , X_{n}$ are i.i.d continuous random variables with mean $\mu$ and variance $\sigma^2$, then as $n \rightarrow \infty$, $$ \sqrt{n} \frac{\bar{X} - \mu}{\sigma} $$ will have the standard normal distribution

or

2. If $X_{1}, ... , X_{n}$ are i.i.d continuous random variables with mean $\mu$ and variance $\sigma^2$, then as $n \rightarrow \infty$, $$ \sqrt{n} \frac{\bar{X} - \mu}{\sigma} $$ will have normal distribution $N(0, \sigma^2)$

According to a book: "Introduction to Mathematical Statistics", by Hogg and Craig, the first one is true. But I also see from other sources that the second one is true.

Thanks.

Answer 1

Since $X_i$ are iid, we know that $$\mathrm{var}\left(\sqrt{n}\frac{\bar{X}-\mu}{\sigma}\right)=\frac{n}{\sigma^2}\mathrm{var}\left(\bar{X}-\mu\right)=\frac{n}{\sigma^2}\mathrm{var}\left(\frac{1}{n}\sum_{i=1}^nX_i\right)=\frac{n}{\sigma^2}\frac{1}{n^2}(n\sigma^2)=1$$ so the limit is a standard normal.

Version 1 is correct, version 2 is wrong.