To prove the equivalence of the induction principles, why does it seems we assume what we want to prove?

Question

I've been trying to understand the proof of equivalence of induction, from Gunderson's Handbook of Mathematical Induction:

The proof given in the book is as follows:

I have looked at other sources, for example, Stillwell's Elements of Algebra:

There is something I am confused: What are they doing - logically - in there? In both proofs, it seems they are trying to prove the second form of induction from the first, that is: $weak \implies strong$, what confuses me is that (I guess) they both suppose that what they are trying to prove is true. I guess the most clear proof I've found was in an article by A. Schach and it is as follows:

And yet, it seems to be assuming what was to be proved again. What is happening?

Answer 1

There is something I am confused: What are they doing - logically - in there? In both proofs, it seems they are trying to prove the second form of induction from the first, that is: weak⟹strong, what confuses me is that (I guess) they both suppose that what they are trying to prove is true.

Names are deceiving. Strong induction is NOT a stronger form of induction. In fact, strong induction and induction (and also the well ordering of $\mathbb{N}$) are equivalent. This means they imply each other. Your quoted proofs are proving that normal induction implies strong induction. There should be a proof of the converse somewhere else in the book.