For a converging(the radius of convergence $0 < r < \infty$) complex power series $\displaystyle \sum a_n z^n$ prove that there exists a positive number $C$ such that if $\displaystyle A > \frac{1}{r}$ then $$\lvert a_n \rvert \le CA^n, \forall n \in \mathbb{N}$$
So I tried this: We know that $$\frac{1}{r} = \limsup_n\lvert a_n \rvert^{1/n}$$ Let's construct a series $\displaystyle \sum_n \left(\frac{1}{A}\right)^n\lvert a_n \rvert$ then by the root test, it converges because $$\limsup_n \left(\frac{1}{A}\right) \lvert a_n \rvert^{1/n} < 1$$ So the sequence converges, then it is bounded. Then there exists a positive number $C$ such that $$\lvert a_n \rvert \le CA^n, \forall n \in \mathbb{N}$$ I think my proof is too nasty and it proves existence depending on $A$. How should I prove this?