Let $(X_n)$ be a sequence of i.i.d. random variables with $E[X_1]=1$ and $Var[X_1]=\sigma^2$. Show that $\frac{2}{\sigma}(\sqrt{T_n}-\sqrt{n})\xrightarrow {d} N(0,1)$ where $T_n=\Sigma_{i=1}^{n}X_i$
I don't really know how to even start this proof, but I think the central limit theorem can be used. However, I don't know which version of the theorem I should use.
Write $T_n=S_n+n$. The CLT states that $\frac{S_n}{\sigma\sqrt{n}} \xrightarrow {d} N(0,1)$ and the strong LLN gives that $S_n=(1+o(1))n$ a.s. as $n \to \infty$.
Observe that $$\frac{2}{\sigma}(\sqrt{T_n}-\sqrt{n})=\frac{2}{\sigma}\frac{T_n-n}{\sqrt{T_n}+\sqrt{n}}=\frac{2}{\sigma}\frac{S_n}{(2+o(1))\sqrt{n}} \,, $$ and the conclusion follows.