Showing 2 Distributions are the Same

Let $\phi_X(u)$ indicate the characteristic function of the r.v. $X$ in the following.

Moreover let $\mathbf{X}=[X_1,X_2,...,X_n]$. Let $\mathbf{A}=\{A_j\}$, where $A_j=\frac{n!}{j},$ $1\le j\le n$.

Using independence of the $X_j$ and the scaling property of the characteristic function you have: $$ \begin{align} \phi_{U_n}(u)&=\phi_{\sum_{j=1}^n \frac{X_j}{j}}(u)\\ &=\phi_{\frac{\mathbf{A}^T\mathbf{X}}{n!}}(u)\\ &=\phi_{\mathbf{A}^T\mathbf{X}}\left(\frac{u}{n!}\right)\\ &=\phi_{\mathbf{X}}\left(\frac{\mathbf{A}^Tu}{n!}\right)\\ &=\left(\phi_{X_1}\left(\frac{A_1}{n!}u\right)\right)^n\\ &=\phi_{Y_n}\left(u\right),\\ \end{align}$$

and the characteristic function characterizes the probability.