Another way to understand inequality is to only admit that the endpoint is also a rational number interval. Of course, this difference has little effect on this problem.

The bigger problem of this problem is that "any of the three basic lemmas proved above is no longer valid". In fact, his original intention is that the three lemmas proved above are still valid, or we can find corresponding examples.

Generally speaking, Zoroach is not careful in writing books, and there are many logical mistakes in the book. So if he is a beginner, I suggest not to use this book as the only teaching material, but to study the ideas in it as a reference book.