Integral of an analytic complex function inside a closed disk

Question

I am trying to solve an exercise from Busam, Freitag Complex Analysis 2nd ed. which also has a provided solution but I guess I did not understand it.

Here is the problem:

and the provided solution:

Why does $f_r$ converges to $f$ uniformly and how it proves that the integral vanishes?

Answer 1

$f$ is uniformly continuous on $\overline{\Bbb E}$, as a continuous function on a compact set. It follows that $$ \oint_{|\zeta| = 1} f(\zeta) \, d\zeta = \oint_{|\zeta| = 1} \lim_{n \to \infty} f\left((1-\frac 1n)\zeta\right) \, d\zeta = \lim_{n \to \infty} \oint_{|\zeta| = 1} f\left((1-\frac 1n)\zeta\right) \, d\zeta $$ and all integrals on the right are zero according to Cauchy's integral theorem.