Let $X_1, X_2, ....$ be iid RV with mean 0, variance 1, and $E(X_i^4)$ is finite, Show that the limiting distribution of $Z_n = \sqrt{n} \frac{X_1X_2 + X_3X_4 +..... + X_{2n-1}X_{2n}}{X_1^2+X_2^2 + .... + X_{2n}^2}$ converges to N(0, 1) in distribution.
I think this problem uses CLT to prove. But I'm just stuck on where to start. Any help will be appreciated.