Lesson 4

From the theorem we have just proved, it follows that the sets \(\mathbb{Z}_{>0}^{2}\) (pairs of positive integers) and \(\mathbb{Z}_{>0}^{3}\) (triplets) are countable. Moreover, for every \(k\in\mathbb{Z}_{>0}\), the set \(\mathbb{Z}_{>0}^{k}\) is countable. In fact, the set of all finite sequences of integers is countable. This set is simply \[\bigcup_{k=1}^{\infty}\mathbb{Z}^{k}.\]