Question: Let A(n) be the statement: "Any collection of n professors in one room all have the same income". Consider the following "induction argument": A(1) is obviously true. Suppose A(k) is true for some natural number k. We will then prove that A(k+1)is true. So take any collection of k + 1 professors in one room and send one of them outside. The remaining k professors all have the same income by the induction hypothesis. Bring the professor back inside and send another outside instead. Again the remaining professors will have the same income. But then all the k + 1 professors will have the same income. By induction, this proves that all n professors have the same income. What is wrong with this argument?