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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 [itex]\mathbb{R}[/itex] and suppose that

[itex]\mbox{sup }S\notin S[/itex]. Prove that there is a non decreasing sequence

[itex](s_n)[/itex] of points in S such that [itex]\lim s_n =\mbox{sup }S [/itex].

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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