. Verify that the Mathematical Induction is valid.

The validity of Mathematical Induction can be proved by using well-ordering property.

Assume that P(1) is true and P(k)→P(k+1) is also true. If the proof using mathematical induction is not valid then P(n) is true for all positive integers would be false.

Let the set of positive integers for which P(n) is false be T where T is a non-empty set. Since there is at least one element in T, by well-ordering property, T has a least element. Let the least element be m. Then m cannot be 1, since P(1) is assumed as true. Hence, m must be greater than 1.

So, m-1 is a positive integer other than 1, which is also not in T as m is the least element in T. This shows P(m-1) is true as m-1 is not in T.

Also by our assumption, P(m-1)→P(m), so P(m) must be true to make the implication true. T. Hence, P(n) must be true for all positive integer n.

Therefore, mathematical induction is valid.