**Question:**

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> Prove that every nonempty open subset $G$ of $\mathbb{R}^n$ can be expressed as a countable union of nonoverlapping closed rectangles, $G= \bigcup\_{k=1}^\infty I\_k$.
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**Attempt:**
My idea is to tile $\mathbb{R}^n$ with cubes of side 1. Then discard the cubes that do not intersect with $G$. Then repeat. However, I'm not sure if this is the right approach, since it may only approximate $G$. I also would be unsure of how to formalize such an approach. Alternative ideas would be appreciated.

Here's some additional context since there's quite a bit of discussion on the problem. The problem as originally written in Chapter 2 of Lebesgue Integration on Euclidean Space by Frank Jones reads:
![enter image description here](https://i.stack.imgur.com/9GVhq.png)

Of course, we need to know what a special rectangle is. This is defined as follows (essentially just a closed rectangle):
![enter image description here](https://i.stack.imgur.com/T2WUH.png)