How is it true that $\vert e^{iw(r)} - 1 \vert \leq \min\{\vert w(r) \vert, 2\}$?

Question

I'm reading a thesis where the author uses this inequality (page 33) to prove a lemma:

I know $\vert e^{iw(r)} - 1 \vert = \sqrt{2-2\cos(w(r))}$, which is at least $0$ and at most $2$, but how can I conclude that $\vert e^{iw(r)} - 1 \vert \leq \min\{\vert w(r) \vert, 2\}$? I've included the link to the thesis in case there is context missing. I've been stuck with this for a while now.

Answer 1

Draw a picture. $|e^{iw}-1|$ is a length of side $[e^{iw}e^{i0}]$ of a triangle $\Delta e^{iw} e^{i0} o$ and $w$ is length of arc. If $w>2$, then $|e^{iw}-1| \leq 2$.