I am reading this note online and I am having trouble understanding the proof of theorem 20, Caretheodory's theorem. The author writes
"...Since $\int_0^2 \frac{dr}{r}$ diverges near $r=0$, we conclude from the pigeonhole principle that there exists a sequence of radii $0 < r_n < 2$ decreasing to zero such that $$\displaystyle r_n^2 \int_{0}^{2\pi} 1_{D(0,1)}(\zeta+r_ne^{i\theta}) |\phi'( \zeta + r_n e^{i\theta} )|^2\ d \theta \rightarrow 0$$ and hence by Cauchy-Schwarz $$\displaystyle r_n \int_{0}^{2\pi} 1_{D(0,1)}(\zeta+r_ne^{i\theta}) |\phi'( \zeta + r_n e^{i\theta} )|\ d \theta \rightarrow 0.$$ If we let $C_n$ denote the circular arc $\{ \zeta + r_n e^{i\theta}: 0 \leq \theta \leq 2\pi \} \cap D(0,1)$, we conclude from this and the triangle inequality (and chain rule) that $\phi(C_n)$ is a rectifiable curve with length going to zero as $n \rightarrow \infty$."
I do not understand how pigeonhole principle gets applied (i.e. what are the "pigeons" and "pigeonholes"). Someone suggested to me that the integrand is of order $o(\frac{1}{r^2})$, why is that the case? Also, the Cauchy-Schwartz inequality gets applied to the function $f := r_n$ and $g$ as the integrand, right? Also, I don't understand how we can conclude that "from this and the triangle inequality (and chain rule) that $\phi(C_n)$ is a rectifiable curve with length going to zero as $n \rightarrow \infty$." Thank you for your help.