The Real And Complex Number Systems

then it is clear that
So, F n = x n for all n.
F 1 = 1, F 2 = 1, and F n+1 = F n + F n−1 for n > 1.
Remark: The study of the Fibonacci numbers is important; the reader
can see the book, Fibonacci and Lucas Numbers with Applications
by Koshy and Thomas.
1.6 Prove that every nonempty set of positive integers contains a smallest
member. This is called the well–ordering Principle.
Proof: Given (φ ≠) S (⊆ N) , we prove that if S contains an integer
k, then S contains the smallest member. We prove it by Mathematical
Induction of second form as follows.
As k = 1, it trivially holds. Assume that as k = 1, 2, ..., m holds, consider
as k = m + 1 as follows. In order to show it, we consider two cases.
(1) If there is a member s ∈ S such that s < m + 1, then by Induction
hypothesis, we have proved it.
(2) If every s ∈ S, s ≥ m + 1, then m + 1 is the smallest member.
Hence, by Mathematical Induction, we complete it.
Remark: We give a fundamental result to help the reader get more. We
will prove the followings are equivalent:
(A. Well–ordering Principle) every nonempty set of positive integers
contains a smallest member.
(B. Mathematical Induction of first form) Suppose that S (⊆ N) ,
if S satisfies that
Then S = N.
(1). 1 in S
(2). As k ∈ S, then k + 1 ∈ S.
(C. Mathematical Induction of second form) Suppose that S (⊆ N) ,
if S satisfies that
(1). 1 in S
(2). As 1, ..., k ∈ S, then k + 1 ∈ S.
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