Violation of Induction Hypothesis

Definition of an inductive set A subset A of the real numbers is said to be inductive if it contains 1, and for every element x in A, the number x + 1 is also in A.
Here we show that a product set is not inductive. We shows that if C is subset of natural numbers. If 1 is in C, and let max(C) be the largest element of C, there is an element n < max(C) that does not belong to C.
One counter example would be sufficient to prove it. Let A = {1, 2}. A × A = {1,2, 4}. One can see that 3 is not in the set.