## questions on inductive definitions in a proof

Hi

I was trying to solve the following problem from Kenneth Ross's Elementary Analysis book.
here is the problem.

Let S be a bounded nonempty subset of $\mathbb{R}$ and suppose that
$\mbox{sup }S\notin S$. Prove that there is a non decreasing sequence
$(s_n)$ of points in S such that $\lim s_n =\mbox{sup }S$.

Now the author has provided the solution at back of the book. I have attached the snapshot of the proof. I am trying to understand it. He is using induction here in the proof. Now in induction, we usually have a statement P(n) , which depends upon the natutal number n. And then we use either weak or strong induction. So what would be P(n) in his proof. I am trying to understand the logical structure of the proof. Thats why I decided to post in this part of PF.

thanks
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 It's induction in the sense that given the n-1 term he can construct the nth term. He starts with the 1st term and shows you how to construct the 2nd term. He could then, just as well, have said "proceeding in this manner". Notice that below he just says "therefore the construction continues".
 Sorry, maybe I didn't answer your question. He shows that the first term exists. Then he shows that given that the n-1 term exists then the nth term exists by his construction. Therefore all terms exist.