analysis - Show that $sum_{n=0}^infty a_n z^n$ converges $forall zinmathbb{C}.$

Assume that $\sum_{n=0}^\infty b_n z^n$ converges $\forall z\in\mathbb{C}.$ Let $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|$ exists. Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall z\in\mathbb{C}.$

Proof:

Since $\sum_{n=0}^\infty b_n z^n$ converges $\forall z\in\mathbb{C},$ the radius of convergence $R_{(b_n)}=\infty, \mbox{ i.e. } \limsup_{n\rightarrow\infty} b_n = 0.$ As $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|$ exists, we can write $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|=\limsup_{n\rightarrow\infty}|\frac{a_n}{b_m}|.$ I am stuck here.

I know that to show $\sum_{n=0}^\infty a_n z^n$ converges $\forall z\in\mathbb{C},$ I need to prove that $\limsup_{n\rightarrow\infty}a_n=0.$ Any suggestions?

Answer

Since the limit $\lim_{n\to \infty}\left|\frac{a_n}{b_n}\right|$ exists it follows that the sequence $\left|\frac{a_n}{b_n}\right|$ is bounded.
Therefore there exists a positive number $M$ such that $\left|a_n\right|\leq M\left|b_n\right|$ for all $n\in\mathbb N$.
We conclude that $\left|a_nz^n\right|\leq M\left|b_nz^n\right|$ for all $n\in\mathbb N$ and $z\in\mathbb C$ and the result follows.