By Abraham A. Fraenkel

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**Example text**

We proceed t o an essentially different instance of a denumerable set. Between two consecutive integers n and n + 1, there are infinitely many rational numbers See p. 38. -3 -* ’ , I I I I I I I an arbitrary row as well as an arbitrary column by 0; CH. I, 0 31 49 DENUMERABLE SETS (- 1 / 1 ) , etc. Arrange the lattice points in the same order in which they succeed each other on the given route. (In arithmetic and analysis one calls such a rearrangement of a doubly infinite sequence into a simply infinite sequence, the diagonal method of Cauchy, for it was Cauchy who used this method in the theory of infinite series; cf.

CONCEPT O F CARDINAL NUMBER [CH. Any subset of a denumerable set D is either finite or denumerable. Proof. One could simply say, any subset is either finite or infinite, and in the latter case denumerable because of theorem 1. D. It may, however, be useful to illuminate the constructive character of the proof by accomplishing it in a more detailed way, which also applies to theorem 1. Denote D again by {al, d,, . , d,, . }, using a certain representation between D and the set of all natural numbers ; let Do be any subset of D.

Any subset of a denumerable set D is either finite or denumerable. Proof. One could simply say, any subset is either finite or infinite, and in the latter case denumerable because of theorem 1. D. It may, however, be useful to illuminate the constructive character of the proof by accomplishing it in a more detailed way, which also applies to theorem 1. Denote D again by {al, d,, . , d,, . }, using a certain representation between D and the set of all natural numbers ; let Do be any subset of D.