Q:

Question concerning proof of the Cauchy Integral Formula on the Wolfram website

I was reading through the proof of the Cauchy Integral Formula here. I do not understand 
how the transition is made from equation (8) to equation (9). While taking the limit as $r\to 0$, doesn't the closed curve $\gamma_r$ also vanish? So, by then the closed curve $\gamma_r$ around $z_0$ is degenerate(a point), I think.
Can you please explain what is going on?
Thank you.

A:

The integrate $\oint_{\gamma_r} f(z_0 + r \mathrm{e}^{i \theta}) i \mathrm{d} \theta$ really means $\int_0^{2\pi} f( z_0 + r(\theta) \mathrm{e}^{i \theta}) i \mathrm{d} \theta$, where $r(\theta)$ is bounded by $r$, i.e. $\sup_{0\le \theta < 2\pi} r(\theta) \le r$.
With this in mind, the eq. (9) will follow by continuity of $f(z)$.

