Let $P$ be a packing in $\mathbb{R}^2$ consisting of infinitely many unit disks. Is it possible for $P$ to have density $0$?
For clarification:
A packing $P$ of $D \subset$ $\mathbb{R}^2$ is a set of unit disks with pairwise disjoint interior. Density is defined as
$$\rho(P, D) = \frac{\text{Area covered by the disks of P}}{\text{Area of }D}$$
In case of infinite packings we have to be a bit more careful: Let $P_D \subset P$ such that each disk in $P_D$ is completely contained in $D$. We further define $Box(n) = [-n,n] \times [-n,n]$. The density of $P$ is then given by
$$\rho(P) = \lim_{n \rightarrow \infty} sup \ \rho(P_{Box(n)},Box(n))$$
I suppose that the claim is true, but I can not find an example. Could you give me a hint?