$(a^n-b^n)$ bounded implies $a=b$.

Question

Let $a,b\in\mathbb{C}$ such that $|a|=|b|>1$. If the sequence $(a^n-b^n)$ is bounded, then $a=b$.
Tried using polar coordinates but it didn't realy work out. Also, the formular for the difference of n-th powers doensn't seem very useful here. Any tips?

Answer 1

As $|a|=|b|, a=be^{i\theta}$ for some real $\theta$. $a^n-b^n=a^n(1-e^{in\theta})$ The factor in parentheses cannot always be small unless $\theta=0$