By the well-ordering property S has a least element say m. We first establish that the proposition P n is true for the lowest possible value of the positive integer n. P k P k 1 P k P k 1 If you can do that you have used mathematical induction to prove that the property P P is true for any element and therefore every element in the infinite set.
1 k 1 log k 1 And work on that until you show that it is less than or equal to k 2 2.
The key element in making the Induction step is to prove this chain of equalities and inequalities. P k P k 1 P k P k 1 If you can do that you have used mathematical induction to prove that the property P P is true for any element and therefore every element in the infinite set. Suppose that P1 holds and Pk Pk 1 is true for all positive integers k. This is the case when grows much faster than.