and the following is a hint from same book
I’d like to know why we should consider $f_n(z)$ instead of $f(z)$. Is there an advantage of considering such auxiliary function?
And also on the circle $C$, at $\zeta =1$, we don ‘t know whether $f$ is analytic or not, but the author says that $ |F_n(z)| \lt \varepsilon$ for all $\zeta \in C$.
Should it be $C-{1}$ instead of $C$?
Thank you for your comment in advance.
MMP is applied to the circle $C$. By hypothesis there exists $\delta >0$ such that $\zeta \in C$ and $|\zeta -1| <\delta$ implies $|F(z)| <\epsilon$. Now consider points $\zeta \in C$ with $|\zeta -1| \geq \delta$. If you draw a picture of the two circles you will notice immediately that $|\zeta|$ stays away from $1$. Hence $|\zeta|^{n} \to 0$ uniformly for $\zeta \in C$ and $|\zeta -1| \geq \delta$ Hence we can make $|F(\zeta)| <\epsilon$ by choosing $n$ sufficiently large. The reason for considering $F$ should now be clear.