# Questions on inductive definitions in a proof

by issacnewton
Tags: definitions, inductive, proof
 P: 610 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 Attached Thumbnails