Let $Y_1, Y_1, \dots$ be independent, identically distributed random variables with the uniform[0,1] distribution and let $X_k=k\cdot Y_k,\; S_n=X_1+X_2+ \dots +X_n$. How to prove that $\frac{S_n}{\frac{n^2}{4}} \,{\buildrel \text{weakly} \over \to_{n \to \infty}}\, 1 $ and $\frac{S_n-\frac{n^2}{4}}{\frac16n^{\frac32}} \,{\buildrel \text{weakly} \over \to_{n \to \infty}}\, \mathcal{N}(0,1).$
Any help would be really appreciated!