I have a problem let s2 = {x in R : x > 0}. Does s2 have lower bound, upper bound? Does inf(s2) and sup(s2) exist?
I understand the that the lower bound is 0 while there is no upper bound. I think I can prove that there is no upper bound since if I say let v be an upper bound so max(0, v+1) > v and max(5, v+1) is in S, which cannot happen. But I am stuck on proving that a) inf(s2) exists, and b) sup(s2) does not exists. I am not sure how to prove these two things.
Here is a sketch that maybe you can use in your proof:
We have $S_2 = \{x\in\mathbb{R} \; | \; x>0\}$. What is a lower bound for $S_2$. Is any negative number a lower bound? What about $0$? Now let's consider what might be the greatest lower bound. Can any positive number a lower bound? Why can't $a>0$ be a lower bound for $S_2$?