Doubts In Proof of Holomorphic function defined on Integral

Question

I was reading Complex Analysis Stein And Shakarchi .
In that I had Doubts regarding One step in theorem 5.4 page 56 as follows

I do not understand Why $|f_n(z)-f(z)|=$ is written in Yellow shaded part
Any Help will be appreciated

Answer 1

Because\begin{align}f_n(z)-f(z)&=\frac1n\sum_{k=1}^nF\left(z,\frac kn\right)-\int_0^1F(z,s)\,\mathrm ds\\&=\left(\sum_{k=1}^n\int_{\frac{k-1}n}^{\frac kn}F\left(z,\frac kn\right)\,\mathrm ds\right)-\left(\sum_{k=1}^n\int_{\frac{k-1}n}^{\frac kn}F(z,s)\,\mathrm ds\right)\\&=\sum_{k=1}^n\left(\int_{\frac{k-1}n}^{\frac kn}F\left(z,\frac kn\right)-\int_{\frac{k-1}n}^{\frac kn}F(z,s)\right)\,\mathrm ds.\end{align}