Let $U'_{n,i}, 1\leq i \leq r_n$, be independent random variables having the same distributions as $U_{n,i},1 \leq i\leq r_n$, and assume that $\sup_t\mid \varphi_{U_{n,i}+U_{n,j}}(t) -\varphi_{U'_{n,i}+U'_{n,j}}(t)\mid\leq C\alpha_n$, for some constant $C$, for any $i\neq j: i,j=1,\dotsc,r_n$, where $\varphi_X$ denotes the characteristic function of $X$.
I want to show that for the summation, a uniform bound also holds, i.e., $\sup_t\mid \varphi_{\sum_{i=1}^{r_n}U_{n,i}}(t) -\varphi_{\sum_{i=1}^{r_n}U'_{n,i}}(t)\mid\leq r_nC\alpha_n$. If $Cr_n\alpha_n\to0$ as $n\to \infty$, then $\sum_{i=1}^{r_n}U_{n,i}\to\sum_{i=1}^{r_n}U'_{n,i}$ in distribution, as far as I understand. My question is:
How can I show this uniform bound for the summation using, e.g., induction?