I'm currently noodling through a proof as to why a Gamma distribution of $Gamma(2, \frac{1}{2} )$ converges as per the Central Limit theorem: $$ \sqrt{n}( \overline{X}_{n} - \mu) \rightarrow_{d} N(0, \sigma^{2}$$
Also, from the Gamma Distribution I know that: $$E(X) = \alpha\beta = 1$$ $$Var(X) = \alpha\beta^{2} = 1/2$$ When I plug these value in I get: $ \sqrt{n}( \overline{X}_{n} - 1) \rightarrow_{d} N(0, \frac{1}{2})$.
My understanding starts to fall apart when I use the formula :
$$Z_{n} = \frac{\overline{X}_{n}-\mu}{\sigma/\sqrt{n}}$$ I get: $Z_{n}=\sqrt{2n}(\overline{X}_{n}-1)$ Which kind of looks like $ \sqrt{n}( \overline{X}_{n} - 1)$ with the $n$ being scaled by a factor of $2$. I'm fully aware that $\lim_{n \rightarrow \infty} \sqrt{2n}=\lim_{n \rightarrow \infty} \sqrt{n}$. So maybe this is the fact I'm supposed to use.
But I'd like to have a clear understanding of how to precisely prove the convergence rather than plugging in values and saying it's done.
Thank you for reading.