Answer
Have to Show that
--if the statement P(n) is true for infinitely many positive integers
-n and P(n + 1) → P(n) is true
-for all positive integers n, then P(n) is true for all positive integers
n.
Work Step by Step
--Let:,
-for a proof by contradiction,
- that there is some positive integer n such that
-P(n) is not true.
--Let
- m be the smallest positive integer greater than n for which P(m) is true;
- we know that such an m exists
because P(m) is true for infinitely many values of m.
- But we know that P(m)→P(m−1), so P(m−1) is also true.
- Thus,
-(m − 1) cannot be greater than n, so (m − 1) = n and P(n) is in
fact true.
---This contradiction shows that P(n) is true for all n.