MATHEMATICAL INDUCTION
Principle of Mathematical Induction:
Let S(1) , S(2) , S(3) , ... be a sequence of statements satisfying
i) S(1) is true, and
ii) For each positive integer k , S(k) S(k+1).
Then S(n) is true for every positive integer n .
Principle of Mathematical Induction - Generalized:
Let S(j) , S(j+1) , S(j+2) , ... be a sequence of statements satisfying
i) S(j) is true, and
ii) For each positive integer k j , S(k) S(k+1).
Then S(n) is true for every positive integer n j .
Principle of Strong Induction:
Let S(j) , S(j+1) , S(j+2) , ... be a sequence of statements satisfying
i) S(j) is true, and
ii) For each positive integer k j,
If S(j) , S(j+1) , ... , S(k) are all true, then S(k+1) is true.
Then S(n) is true for every positive integer n j .
Well Ordering Principle for Natural Numbers:
Every non-empty collection of positive integers contains a least element.