filling a square with circles
I found this problem on Charles Chapman book, Real Mathematical Analysis:
Given epsilon>0, prove that finitely many disjoint circles can be draw inside the unit square so that the total area of the discs exceeds 1 - epsilon.
The first idea that comes to my mind is to try to fill the square with ever smaller disks of the same size. But the problem with this approach is that the total area covered by the disks is always the same! It will not change as you decrease the size of the disks.
So all the disks cannot be of the same size. I need a new strategy. Any ideas?
By the way, this is all in the plane. Maybe it can be extended to more dimensions, but I will be satisfied with a solution in 2D.