Problem:

Suppose that A is not an empty set and is bounded below, let B be the set of all lower bounds of A.

1.) Show that B is not an empty set

By the definition of bounded below, there exists an element b in the set B where b <= a for every a in A. Thus, B is not an empty set.

2.) Show that B is bounded above

I'm a bit lost here; I believe that I have to prove that there exists some number n such that for every b in the set B, m has to be greater than or equal to b. But I'm not sure how to prove this.

3.) Show that that sup B is the greatest lower bound of A.

In essence, I'm trying to prove that sup B = inf A. Which intuitively I understand, but I'm not sure how to approach showing this.

I am really struggling with conceptually following proofs of least upper bounds. Any help would be greatly appreciated!

Thanks for your time.