I have the following problem.
We have $R\subset \Bbb{C}$ a rectangle and $\{f_n\}$ a collection of analytic functions which converges uniformly to $f$. After a long computation we got to the following point $$\int_{\partial R} |f(z)-f_n(z)|dz$$ Then our prof wrote the following inequality:$$\int_{\partial R} |f(z)-f_n(z)|dz\leq max_{z\in \partial R}|f_n(z)-f(z)|\cdot\int_{\partial R} |dz|$$
And I somehow don't see why one can take the maximum outside and leave the rest.
Could maybe someone explain this to me?
Thanks for your help