Note that

Let be a well-ordered set. A subset of is said to be inductive if for every

Theorem (The principal of transitive induction). If J is a well-ordered set and is an inductive subset of , then .

Can someone help me get started on proving this. I am trying to show that [tex] J \subseteq J_0 [/tex] by choosing assuming that and showing that there is a contradiction. Is this a good approach and do you have any hints that I can use to get anywhere?

Ok so I made some progress, but I am not sure if I am moving in the right direction.

Proof: Choose and assume that Then

If then consider

Then this set has a least-element, call it It follows that

This is a contradiction.

If for this case, I can not find any kind of contradiction. Any suggestions?