Question

Prove that the set n of natural numbers is an infinite set

Answer 1

Suppose for the sake of contradiction that the set of natural numbers is finite. Then there exists a maximum element mm. But m+1=nm+1=n is also a natural number and n>mn>m. This contradicts the maximality of mm, so our original assumption was false, and hence set of natural numbers is not finite but rather infinite.

The induction argument fails because it shows P(n)P(n) is finite for every natural number nn, but it does not show that P(∞)P(∞) is finite because ∞∞ is not a natural number.