TITLE: Covering the plane by squares!
QUESTION [6 upvotes]: $K_n$ 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.

REPLY [3 votes]: 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.)