I reading a proof of a preliminary version of Cauchy's theorem, on page 109 of Ahlfor's Complex Analysis.
He's proving that if the function $f(z)$ is analytic on a rectangle $R$, then $\int_{\partial R}f(z)dz=0$.
His notation is that $\eta(R)=\int_{\partial R}f(z)dz$. He breaks $R$ into four congruent rectangles, so $\eta(R)=\eta(R^{(1)})+\cdots+\eta(R^{(4)})$, and chooses some rectangle, denote it $R_1$ such that $|\eta(R^{(k)}|\geq\frac{1}{4}|\eta(R)|$. Repeating this process the rectangles converge to some point $z^*$.
Then given $\epsilon$, I know you can chooise $\delta>0$ such that for $|z-z^*|<\delta$, \left|\frac{f(z)-f(z^*)}{z-z^*}-f'(z^*)\right|<\epsilon or |f(z)-f(z^*)-(z-z^*)f'(z^*)|<\epsilon|z-z^*|.
Since $\int_{\partial R_n}dz=0$ and $\int_{\partial R_n}zdz=0$, he writes \eta(R_n)=\int_{\partial R_n}[f(z)-f(z^*)-(z-z^*)f'(z^*)]dz.
How exactly is this final equality justified? I don't see where it's coming fr4om, thanks.