1. Introduction to the theory of infinite sets. Theorem 1.1. N is infinite. Summary: 1. Introduction to the theory of infinite sets. Theorem 1.1. N is infinite. Proof. We have N N {0} by means of S. Thus N cannot be finite since, as we have already shown, no finite set is equipotent with a proper subset. Theorem 1.2. Suppose A N and A is infinite. Then A N. Proof. We do this by defining a function by induction. Let G = {g : for some n N, g : I(n) A} and define G : G A by letting G(g) be the least element of A rng g for g G. One obtains a function f : N A such that f(n) = G(f|I(n)) for each n N. We leave to the reader the straightforward verification that f is univalent with range equal to A. Remark 1.1. Note that in the above proof f carries n to the n-th least element of A. Alternatively, we could use the ordering on N to induce a well ordering on A and then use our previous results about well ordered sets. Theorem 1.3. Nn N for any n N+ . Proof. The statement is trivially true when n = 1 and will follow by induction on n if we can show it holds for n = 2. We now show that N N × N. For each n N we let L(n) = {(l, m) N × N : l + m = n} and we let M(n) = {L(m) : m < n}. Note that L(n) is Collections: Mathematics