So I'm trying to prove a result in complex dynamics, but I came across the following lower bound $$|e^{2\pi i\omega q}-1|=|e^{2\pi i\omega q}-e^{2\pi i p}|=|e^{2\pi i(q\omega -p)}-1|\geq K|q\omega-p|$$ where $p,q\in\mathbb{Z}$. I don't really know what it is being used in the last two steps and although it seems quite elementary, could anyone explain it to me?
Cheers!
I will start by expanding the second step more clearly. $$|e^{2\pi i\omega q }-e^{2\pi i p}|=\bigg|\frac{e^{2\pi i\omega q }}{1}-\frac{e^{2\pi i p}}{1}\bigg|=\bigg|\frac{e^{2\pi i\omega q }}{e^{2\pi i p}}-\frac{e^{2\pi i p}}{e^{2\pi i p}}\bigg|=|e^{2\pi i(\omega q-p)}-1|$$ As for the final step it is saying for every fixed pair $p,q \in \mathbb{Z}$ there is a constant $K$ making the inequality hold which should be clear as $|e^{2\pi i(\omega q-p)}-1|$ grows at an exponential rate and $K|\omega q -p|$ grows at a linear rate.