## Complete proof of PAC learning of axis-aligned rectangles

I have already read PAC learning of axis-aligned rectangles and understand every other part of the example.

From Foundations of Machine Learning by Mohri, 2nd ed., p. 13 (book) or p. 30 (PDF), I am struggling to understand the following sentence of Example 2.4, which is apparently the result of a contrapositive argument:

… if $$R(\text{R}_S) > \epsilon$$, then $$\text{R}_S$$ must miss at least one of the regions $$r_i$$, $$i \in [4]$$.

i.e., $$i = 1, 2, 3, 4$$. Could someone please explain why this is the case?

The way I see it is this: given $$\epsilon > 0$$, if $$R(\text{R}_S) > \epsilon$$, then $$\mathbb{P}_{x \sim D}(\text{R}\setminus \text{R}_S) > \epsilon$$. We also know from this stage of the proof that $$\mathbb{P}_{x \sim D}(\text{R}) > \epsilon$$ as well. Beyond this, I’m not sure how the sentence above is reached.