For Inaba’s theorem, how do we know the covering of points is independent?

Question

In this paper on how many points can be covered using unit disks: http://2011.cccg.ca/PDFschedule/papers/paper5.pdf (page 5)

They give a proof of the fact that you can cover $10$ points using unit disks. I understand most of the steps except that they assume that the event of covering one point is independent of covering another. How do we know this is the case?

Edit: there was an inequality sign which I misread as an equality sign (somehow).

Answer 1

I do not think that they assumed such independence. The first equality is using the fact that $\mathbb{P}(A)=1-\mathbb{P}(A^c)$. The second equality is due to De Morgan's law. The "$\geq$" is due to union bound (this does not require independence). Let me know if it answers your question.