QUESTION: I seem to be confused about how transfinite induction is carried out. I have looked at several examples and they seem to follow a procedure consisting of grounding the induction, proving the succession case and then proving the limit case. But I still don't quite understand the point of proving the limit case, and to add to my confusion some authors seem to just prove problems requiring transfinite induction with a base case and an inductive step, completely ignoring the limit case. Could someone please provide some clarifications as well as a step by step example of the proper way of implementing transfinite induction. Thanks.
EDIT: I now see the reasoning for setting up the induction, but take a look at this proof of showing that every vector space has a basis.
Proof: Let V be a vector space. By the well-ordering theorem, there is a well-ordering of the elements of V. Therefore, there is an ordinal $\alpha$ such that the elements of $V$ can be put into one-to-one correspondence with $\alpha.$ For each $\beta<\alpha$, let us write $v_\beta$ for the element of $V$ that corresponds to $\beta.$ Now let $B$ be the set of all $v_\beta$ that do not belong to the linear span of their predecessors. It is clear that $B$ is linearly independent. It also spans, since every $v_\beta$ either belongs to $B$ or is a linear combination of earlier elements of $B$, and every element of $V$ is $v_\beta$ for some $\beta.$
Question 2:I don't see any of the distinct steps of transfinite induction in here, yet it is considered a proof by transfinite induction. Can anyone please explain why this is still considered fine, and can you indicate the limit step.
EDIT 2: It seems that Brian's response has answered both of my questions.