1. ## supremum

Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?

2. Originally Posted by inthequestofproofs
Is the supremum of a closed set always in the closed set if this set is bounded? Does it need to be bounded?
I assume, as your last posts indicated, that you are dealing with closed subsets of $\displaystyle \mathbb{R}$. Then the answer is yes. Notice that $\displaystyle \overline{E}=\left\{x\in\mathbb{R}:d(x,E)=0\right\ }$. But, if $\displaystyle E$ is closed then $\displaystyle \overline{E}=E$. So, let me ask you: what is $\displaystyle d\left(E,\sup\text{ }E\right)$?

(if the supremum exists)

It would be zero, and thus, it is in the closed set

4. Originally Posted by inthequestofproofs
It would be zero, and thus, it is in the closed set

Now prove it

$\displaystyle M=[0,\inf[$, which is a closed subset of $\displaystyle \mathbb{R}$.

Then does the supremum of $\displaystyle M$ exist?

if so then $\displaystyle sup\{M\}=\inf$ but would you say that it is in $\displaystyle M$ ?

What about the empty set? it is both open and closed ?

what is $\displaystyle sup\{\varnothing\}$ ? $\displaystyle - \inf$ ?

6. In order to have a supremum, a set must first have an upper bound. $\displaystyle [0, \infty[$ does not have an upper bound so no supremum.