My textbook says the following:(adsbygoogle = window.adsbygoogle || []).push({});

For a closed interval J_n = [a_n, b_n]

"A nested sequence of rational intervals give rise to a separation of all rational numbers into three classes (A so-called Dedekind Cut). The first class consists of the rational numbers r lying to the left of the intervals J_n for sufficiently large n, or for which r < a_n for almost all n. The second class consists of the rational numbers r contained in all intervals J_n. This class contains at most one number, since the length of the interval J_n shrinks to zero with increasing n. The third class consists of the rational numbers r for which r > b_n for almost all n. It is clear that any number of the first class is less than any of the second class, and that any number of the second class is less than any of the third class. The points a_n themselves are either in the first or second class, and the numbers b_n either in the second or third class.

If the second class is not empty, it consists of a single rational number r. In this case the first class consists of the rational numbers less than r, the third class of the rational numbers greater than r. We say then that the nested sequence of intervals J_n represents the rational number r. For example, the nested sequence of intervals [r - 1/n, r + 1/n] represents the number r.

If the second class is empty, then the nested sequence does not represent a rational number; these nested sequences then serve to represent irrational numbers. The individual intervals [a_n,b_n] of the sequence are for this purpose unimportant; only the separation of the rational numbers into three classes generated by this sequence is essential, telling us where the irrational number fits in among the rational ones."

I don't understand the last paragraph. When they say "if the second class is empty, then the nested sequence does not represent a rational number", do they mean that if the nested sequences do not converge to a rational number, then it must represent an irrational number? Or do they mean that the nested sequence of intervals do not contain any point whatsoever (which I don't believe is possible if the length of each sub-interval tends to zero as n goes to infinity)?

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# Nested sequences of rational intervals

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