let $A$ a nonempty subset of $ \mathbb{R} $ that is bounded above, then $ \bigcup A \in A $??

let $A$ a nonempty subset of $ \mathbb{R} $ that is bounded above, then $
\bigcup A \in A $??

in this question $ \mathbb{R} $ is defined as set of Dedekind cuts on $
\mathbb{Q} $
let $ A $ a nonempty subset of $ \mathbb{R} $ that is bounded above, and $
\bigcup A:=\{x|\exists B \in A(x \in B)\} \in A $
I must proof that $ \bigcup A \in A $
Thanks in advance!!