SCHWEITZER 1985 Problem 10

Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]