I need help proving the following statement:
Let $\{D_i\}$ be a finite colection of disks on $\mathbb{R}^2$, $\lambda$ the Lebesgue measure on $\mathbb{R}^2$. Show that there exists a subcollection of disjoint disks in the collection such that their union $V$ satisfies: $$ \lambda(V)\geq\frac{\lambda \left( \bigcup D_i \right) }{4} . $$
So far the only thing I got is that it suffices to show the theorem for a set of disks that are "clustered" in the sense that there is an ennumeration of the $D_i$'s such that $$D_i\cap D_{i+1}≠\emptyset$$ And that the theorem is obviously true for up to 4 disks.