Question about partition of open sets in $\mathbb{R^n}$

Question

I have to prove that any open set $U \subset \mathbb{R^n}$ is a countable union of disjoint limited rectangles.

I proved that it is a countable union of rectangles, the "expected classical" way, I guess: using rationals. However, how can I make the union disjoint ?

Is it true that any countable union of rectangles in $\mathbb{R^n}$ can be written as a countable disjoint union of rectangles? If not, can you give me a counter-example?

Answer 1

Any open ball in the Eucledian norm is a counterexample if $n>1$.