Covering random points in a unit disk

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let Vn = {X1, X2, ., Xn}, where X1, X2, . are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in Vn that cover all of Vn.