Take a square of side $a$.
Now take finite number of circles $(C^1_1,C^1_2,C^1_3,...,C^1_{n_1 \in \mathbb{N}})$ inside the square such that all of their intersection is empty set. Find its net area and call it $ar(C^1)$.
Now take another finite number of circles $(C^2_1,C^2_2,C^2_3,...,C^2_{n_2 \in \mathbb{N}})$ inside the square such that all of their intersection is empty set. Find its net area and call it $ar(C^2)$.
Keep on doing this and find net area of each possible "collection of finite number of circles inside the square". Then make a set $\gamma=\{ ar(C^1),ar(C^2),ar(C^3),... \}$
Finally what I need to show is:
$$\sup \gamma=a^2 \tag1$$
My heuristic guess:
To prove this, I need to prove the following:
by adding more and more circles inside the square, we can make the net area of all circles as close as we want to $a^2$
This seems true but can anyone give a hint of how to prove the above statement so that I can prove $(1)$?
