Prove that a polyrectangle in an open set has a superset polyrectangle.

Question

Theorem:

Let $S \subseteq \mathbb{R}^n$ be an open set. Let $P \subset S$ be a polyrectangle. Then there exists another polyrectangle $P'$ such that $P \subset P' \subset S$.

A polyrectangle is a union of rectangles. (Sets of the following form) $$\bigcup_j\prod_{k=0}^{m-1}[a_{j, k}, b_{j, k}]$$

I have already proved that any polyrectangle can be expressed as a union of disjoint rectangles, but it is difficult to consider rectangles inside an open set.

Answer 1

HINT

Since $S$ is open and for each $j$ , $$\prod_{k=0}^{m-1}[a_{j, k}, b_{j, k}]\subseteq S$$ we can find $\epsilon_{j,k}>0$ such that $$\prod_{k=0}^{m-1}[a_{j, k}-\epsilon_{j,k}, b_{j, k}+\epsilon_{j,k}]\subseteq S$$