The proof of the validity of mathematical induction is by contradiction and depends upon the axiom of well ordering. That axiom states: Every subset of positive integers contain a first or a least integer. Using that axiom, suppose that the statement fails for some positive integer then it fails for sum first J. We know that J is not 1 because it is true for 1. Therefore, J-1 is positive integer and the statement is true for J-1 because J is the first for which it is not true. However if it is true for J-1 it must be true for (J-1)+1 =J. Thus there is a contradiction.