DE-MO 2019 Problem 3

In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is below another rectangle if there is a line $g$ parallel to the $x$ -axis such that the first rectangle is below $g$ , the second one above $g$ and both rectangles do not touch $g$ .

Similarly, we say that one rectangle is to the right of another rectangle if there is a line $h$ parallel to the $y$ -axis such that the first rectangle is to the right of $h$ , the second one to the left of $h$ and both rectangles do not touch $h$ .

Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$