要如何證明數學歸納法...

Generally, you must assume well-ordering to prove the principle of transfinite induction (they are equivalent). Induction should not be taken for granted; there are sets where induction does not hold!

你可以去看簡明數論
剛開始第一章就有說到數學歸納法
那是利用<<歸納公理>>來證明的
有興趣可以自己買來看~
以下是簡明數論部分的內容:

所謂《歸納公理》內容是
設S是N(正整數集)的一個子集,滿足條件
(1)1屬於S
(2)如果n屬於S 則n+1屬於S 那麼S=N

數學歸納法:
設P(n)是關於自然數n的一個命題
if(1)當n=1時P(1)成立
(2)由P(n)成立推出P(n+1)成立
那麼P(n)對所有自然數成立
<>
設使P(n)成立的所有自然數n所組成的集合是S,S是N的子集
由條件(1)知1屬於S,
由條件(2)知若n屬於S 則n+1屬於S
所以由歸納公理得知S=N
證畢#

I question the validity of this proof. I think it is a mere restatement of the conditions of induction.

Usually, a proof of induction on the natural numbers first considers the set of all members such that P(a) is false, and considers the least element of that set.

亦可參考《數學歸納法縱橫談》九章出版