This is from Rudins principles of mathematical analysis.
First are theorems 3.41 and 3.42 which he uses later.
I assume that 3.41 holds for complex numbers? But what about 3.42?, complex numbers does not have an ordering, and the b's are ordered so I assume it must be for real numbers?
Now he is going to use the 3.42 n a complex power series:
Is this a valid proof? I mean he write c=b, but complex numbers does not have an ordering, but in 3.42 he used an ordering. He also write $z^n=a_n$, but in 3.42, does he not assume that $a_n$ is real?
Is it that it is a power series with real coefficients?, is that what he means? If so there is only the problem that $a_n=z^n$, is this allowed considering $z^n$ may be complex?
All numbers here are complex unless indicated otherwise. When an inequality between two numbers appears in the statement of a theorem, that is an indication that they are assumed real.
The first inequality of 3.42 would be better written without absolute value sign. It is just the triangle inequality combined with the estimate $|A_n|\le M$:
$$\begin{split} &\left|\sum_{n=p}^{q-1} A_n(b_n-b_{n+1})+A_qb_q-A_{p-1}b_q\right| \\& \le \sum_{n=p}^{q-1} |A_n(b_n-b_{n+1})|+|A_qb_q|+|A_{p-1}b_q | \\ &\le \sum_{n=p}^{q-1} M(b_n-b_{n+1}) +M b_q +M b_q \end{split}$$