1. In the book, when talking about poles: (see attachment 1)

It declares "there is no harm to assume $|f(z)| > 1 $ in $0 < |z - z_0| < r_0$". But why there is no harm? I mean I understand there exists a small ball around $z_0$ such that $f(z)$ can be very big, however, you see that $g(z) = \frac{1}{f(z)}$, if there is a point in $0 < |z_1 - z_0| < r_0$ s.t. $f(z_1) = 0$, that means I must find a smaller $r_0' < |z_1 - z_0| \le r_0$ ,s.t., $|f(z)| > 1 $ in $0 < |z - z_0| < r_0'$ and go on with this proof. However, that means the decomposition $\frac{H(z)}{(z-z_0)^m} = f(z)$ is only valid in $0 < |z - z_0| < r_0'$, then what about the part $r_0' \le |z - z_0| < r_0$?

2. (Attachment 2) I cannot understand this part: what does repeat mean? I have no idea how to extend the above argument into the situation where there are several poles in the domain (). I can understand how to do it when there are only several removable singularities though.