# How to interpret this problem involving dyadic squares and unit disc?

1. Feb 10, 2014

### Someone2841

The following problem is from the textbook "Real Mathematical Analysis" by Charles Chapman Pugh.

Given $\epsilon > 0$, show that the unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$, and which intersect each other only along their boundaries.​

I find myself unable to understand what the problem is asking to show. At first glace, I thought it was asking to show that given any $\epsilon$, any collection of dyadic squares that 1) had a total area of greater than $\pi - \epsilon$ and 2) intersected along their boundaries would necessarily be finite. However, this assertion is obviously not true. [Let $\epsilon > \pi$ and consider the infinite collection of dyadic squares $\{[0,\frac{1}{2}]\times[0,\frac{1}{2}],[\frac{1}{2},\frac{3}{4}]\times[0,\frac{1}{4}],[\frac{3}{4},\frac{7}{8}]\times[0,\frac{1}{8}],...\}$]

What am I missing?

Last edited: Feb 10, 2014
2. Feb 10, 2014

### Someone2841

After some additional thought, maybe the problem is actually just asking if given an $\epsilon > 0$, does there exists a finite set of dyadic squares whose area is greater than $\pi - \epsilon$, and which intersect only along their boundaries?