I want to show that
$$\frac{\sum_{k=1}^N X_k}{\sqrt{\sum_{k=1}^N X_k^2}} \overset{N\to\infty}{\to} \mathcal{N}(0,1)\text{ in distribution,}$$
where $X_1,X_2,\ldots$ is a sequence of iid random variables with $\mathbb{E}(X_1)=0$ and $\mathbb{E}(X_1^2) = s < \infty$.
Now I know about the CLT, i. e. $\frac{\sum_{k=1}^N X_k}{\sqrt{sn}}\to\mathcal{N}(0,1)$, and about the proof with characteristic functions, but calculating the CF of this thing seems a bit cumbersome and I think I am missing a more elegant approach.
I would appreciate a hint. TIA
Divide numerator and denominator by $\sqrt {sn}$ and apply Slutsky's theorem. After the division, the numerator converges to an $\mathcal N(0, 1)$ while $$ \sqrt{\sum X_i ^ 2 / sn} \to \sqrt{s / s} = 1, $$ by the SLLN. By Slutsky's theorem the ratio converges to an $\mathcal N(0, 1)$ random variable.