Covering 10 points with unit disks

Question

Claim. Any 10 points in the plane can be covered by disjoint unit disks.

Proof. Consider the hexagonal packing of unit disks, whose density is

$$\delta =\frac{\pi }{2\sqrt{3}}>\mathrm{0.9.}$$

Randomly translate the entire packing. For each fixed point, the probability of being covered is exactly $\delta$. Hence, if $X$ is the number of covered points,

$$\mathbb{E}[X]=10\delta >9.$$

Since $X$ is an integer between $0$ and $10$, there must be some translation for which $X=10$. Thus all 10 points are covered by disjoint unit disks.