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)
I wanted to ask what about the set
$\displaystyle M=[0,\inf[$, 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 $\displaystyle sup\{\varnothing\}$ ? $\displaystyle - \inf$ ?