Let $X_i$ be a sequence of i.i.d. random variables with $E(X_i)=0$ and $Var(X_i)=\sigma^2>0$. Prove that the distributions of $(\sum_{i=1}^{n} X_i)/ (\sqrt(\sum_{i=1}^{n}X_i^2))$ converge weakly to the $N(0,1)$ distribution.
I tried using the Lyapunov central limit theorem and Lindeberg but I couldn't get anywhere. Any help would be greatly appreciated.
By SLLN, we have $$\frac{\sum\limits_{k=1}^n X_k^2}{n}\overset{a.s.}{\rightarrow}\mathbb{E}X_1^2=\sigma^2,$$
as $n\rightarrow\infty$. Together with the classical CLT: $$\frac{\sum\limits_{k=1}^n X_k}{\sqrt{n\sigma^2}}\Rightarrow N(0,1)$$
gives the final result you want. You should note that a.s. convergence is stronger than convergence in distribution so we can insert the first convergence here.