How can I prove that $\inf B=\sup A$?

Question

Let $B$ be the set of all the upper bounds of the non-empty bounded subset $A\subseteq\Bbb R$. Prove that $\inf B=\sup A$.

I divided it into two areas ($\inf B>\sup A$, $\inf B<\sup A$) and tried to show a contradiction. Is it the right approach? How can I prove it?

Consider the definition of the supremum. How does it relate to all of the other upper bounds of a set? — Brian61354270, Mar 21 '19 at 00:54
This has been asked many times before. Please search the site before asking a question. — John Douma, Mar 21 '19 at 01:03

score 1 · Answer 1 · answered Mar 21 '19 at 00:54

1

The following is what we can say:

$\sup A\leq b\;\forall\;b\in B\quad$ (Why?)
Also by definition, $\inf B\leq b\;\forall\;b\in B$. In particular $\sup A\in B$.

Can you complete?

answered Mar 21 '19 at 00:54

Yadati Kiran

1 Answers1