- #1
issacnewton
- 1,000
- 29
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 [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
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