A theorem in my textbook says:
Let $(A, < )$ be a totally ordered set. Set A has a least element and principle of transfinite induction holds in A if and only if A is well ordered.
I understand why you need an assumption that A has a least element to prove left-to-right implication in my textbook proof of this theorem. But I can't find an example of the non-well-ordered set where principle of transfinite induction holds... Obviously, that set doesn't have a least element but that ''hint'' didn't take me far.
So, can someone help me?
EDIT: (Due to Brian M. Scott)
We state principle of transfinite induction as follows:
Let $(A, <)$ be a totally ordered set and $B \subseteq A$ which satisfies:
$ (\forall x \in A) (p_A(x) \subseteq B \implies x \in B) $
Then B = A.
( where $p_A(x) = \{ a \in A : a < x \}$ )