Covering the plane by squares! $K_n$ is a sequence of squares of area $a_n$. Show that if $\sum_{n=1}^\infty a_n=\infty$ then we can arrange the squares $K_n$ to cover $\mathbb{R}^2$.

Comments:
-obviously we can suppose that $a_n\to0$ as $n\to\infty$.
-WLoG we can moreover suppose that $a_1\gt a_2\gt a_3\gt\ldots$.
-That's enough to prove that we can cover unit square in the plane, because unit square is compact.
 A: Solved by CMU Maths Lunch Group. Misha Lavrov was the one who came up with the idea. Basically, the covering strategy is to generalize the covering strategy in $1$-dimensional case.
Lemma Given a sequence of decreasing numbers $a_1, a_2, \dots$ such that $\sum a_i^2 = \infty$. Let $n(i)$ be defined inductively: it is the smallest natural number $m$ such that $$a_{n(1) + \dots + n(i-1)+1} + \dots + a_{n(1) + \dots + n(i-1)+m} \ge 1.$$ If $b_i = a_{n(1)+\dots+n(i)}$, then $\sum b_i = \infty$.
Proof of Lemma Note that $$a_{n(1) + \dots + n(i-1)+1} + \dots + a_{n(1) + \dots + n(i-1)+n(i)-1} < 1$$ and so $$\begin{eqnarray}& & a^2_{n(1) + \dots + n(i-1)+1} + \dots + a^2_{n(1) + \dots + n(i-1)+n(i)-1}  + a^2_{n(1) + \dots + n(i-1)+n(i)} \\ &\leq& b_{i-1}(a_{n(1) + \dots + n(i-1)+1} + \dots + a_{n(1) + \dots + n(i-1)+n(i)-1}) + b_i^2 < b_{i-1} + b_i^2.\end{eqnarray}$$ Summing above inequality over all $i$, we have $$\infty = \sum a_i^2 \leq \sum b_{i-1}+b_i^2 < (a_1 + 1)\sum b_i.$$
QED
We may assume the squares have decreasing length of sides $a_1, a_2, \dots$ and by compactness argument, we only need to cover the unit square. We can pick first $n(1)$ squares to cover a 1 by $b_1$ rectangle and the next $n(2)$ squares to cover a 1 by $b_2$ squares, etc. Lemma says $\sum b_i > 1$, and so we can cover the unit square.
A: Follows immediately from the fact that any finite collection of squares with total area $\geq 4$ can cover a square of area 1. (The proof in the link is a little complicated, but the idea is not. Shrink each square so its size is one of 1, 1/2, 1/4, 1/8, .... Then repeatedly find and combine four squares of the same area to make a larger square.)
