If you are trying to prove that a property P holds for all ordinals then you can apply transfinite induction:
The latter step is often broken down into two cases: the case for nonlimit ordinals (ordinals which have an immediate predecessor), where the usual inductive approach can be applied (show that P(b) implies P(b+1)), and the case for limit ordinals[?], which have no predecessor, and thus cannot be handled by such an argument.
Typically, the case for limit ordinals is approached by noting that a limit ordinal b is (by definition) the union of all ordinals a < b and using this fact to prove P(b) assuming that P(a) holds true for all a < b.
The first step above is actually redundant. If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(b) holds for all b < 0.
An example here would be nice
[I agree that some examples should be added, but it's been a long time since I've looked at this material.]
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