Suppose $\sum_{-\infty}^\infty |A_n|^2$ converges. Show that for each $r\in (0,1)$, the series $\sum_{-\infty}^\infty r^{|n|} A_n e^{inx}$ converges uniformly in $x$.
I know that the series $\sum_{-\infty}^\infty r^{|n|} e^{inx}$ converges uniformly in $x$ by M-test. If we can find $M>0$ such that $|A_n| < M$ for all $n$, then we are done. But I am not sure how to get that? Since $\sum_{-\infty}^\infty |A_n|^2$ converges, then $|A_n|^2$ is bounded. Is that it?