I'm reading a paper on tiling open bounded regions in $\mathbb{R}^2$, and I'm not quite sure about this example of tiling a rectangle with a family of squares, where each square touches a specific portion of the boundary of the rectangle. The algorithm presented is something akin to Euclid's algorithm for finding a greatest common divisor, except in this case, the lengths of the sides can be any positive real numbers, not just natural numbers. In particular, I'm confused by the second case the author presents, where ${Q_k}$ is an infinite set. How do we know this tiling covers all of the interior of the rectangle?
Thanks.
The paper is Tiling Bounded Open Sets with Squares that Touch the Boundary by William T. Trotter.
$S_n$ is the part of the interior not covered by the squares $\{ Q_k \mid k < n \}$ and it is always an open rectangle, where one of the corners is $(b,d)$.
We know that $\displaystyle\lim_{k \to \infty} \lambda(Q_k) = 0$, since $\displaystyle\sum_{k=0}^\infty \lambda(Q_k) \le (b-a)(d-c)$. Whenever $\lambda(Q_k) < \lambda(Q_{k-1})$ it must also be the case that $\lambda(S_k) < \lambda(Q_{k-1})$. It follows that there is a subsequence $\{S_{n_k}\}$ with area converging to 0. This implies that $\displaystyle\bigcap_{k\ge 0} S_{n_k} = \emptyset$.