Set Theory | Notation Question - Bounds of Set

Question

Are here a notation for the set of upper bounds of a set, and is there a notation of the lower bounds of a set?

I need to prove this: $\text{Let}\cases{\text{A be a set}\\ S=supA}$

$\text{Prove:}\\ S=SupA \Leftrightarrow \cases{\fbox{1}\,\forall x \in A \rightarrow x \leq S \\\fbox{2}\, \forall \varepsilon (\varepsilon > 0 \rightarrow \exists x_0 ( x_0 \in A \rightarrow x_0 > S - \varepsilon ) }$

(Just so you'd know what I'm trying to do).

Answer 1

There is no standard notation for the set of all upper (or lower) bounds of a set. But you're free to make up a name for this set! You could just write something like

Let $U$ be the set of all upper bounds of $A$ and $L$ be the set of all lower bounds of $A$.

(By the way, it is unclear why you are referring to $A$ as a "group"--it seems you want to say $A$ is a set instead.)