Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

Question

A "box" is a cartesian product of intervals of the type $[a,b]$

I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite difficult

I am aware that a lot of posts already exists in this direction, for example this one: Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] But it is always about intervals on $\mathbb{R}$ and the proofs are all fairly tough.

Does anyone know if there exists a reference of this proof that is sufficiently easy for beginners in analysis?

Answer 1

Chapter 1 of Stein and Shakarchi's Real Analysis textbook has a similar result, and the proof includes a helpful picture. But I think it's worth taking the time to understand Tao's proof. Dyadic cubes are quite useful.