This is Exercise 1.25.b in Pugh’s Real Mathematical Analysis.
In this post, I showed that the unit disc contains finitely many dyadic squares whose total area is arbitrarily close to the area of the disc.
Now I want to show that we can choose disjoint such dyadic squares.
Consider all the dyadic squares whose total area is arbitrarily close to the area of the disc, as constructed in the aforementioned post. Consider one of those squares and let its area be $A$. Divide it by $16$. Choose one of the smaller dyadic squares that is not in the border. Its area is $A/16$.
Do the same procedure for the fifteen remaining squares. The total area is now $A/16+15A/16^2$. Do this over and over again to get the total area $T=A/16+15A/16^2+15^2A/16^3+…+15^{k-1}A/16^k$ where $k$ is the number of steps. As $k$ tends to infinity $T$ tends to $A$, so we can choose $k$ so large that $T$ is arbitrarily close to $A$.
Is my proof correct?