In: Advanced Math
In this problem we prove that the Strong Induction Principle and
Induction Principle are essentially equiv-
alent via Well-Ordering Principle.
(a) Assume that (i) there is no positive integer less than 1, (ii)
if n is a positive integer, there is no
positive integer between n and n+1, and (iii) the Principle of
Mathematical Induction is true. Prove
the Well-Ordering Principle: If X is a nonempty set of positive
integers, X contains a least element.
(b) Assume the Well-Ordering Principle and prove the Strong
Mathematical Induction Principle.