I'm studying real numbers under Dedekind's cuts construction ($x=\{x\in \mathbb{Q}:x<q \text{ for some } q\in\mathbb{Q\}}$ would be a real number).
The theorem I'm trying to understand is this: for any $\varnothing \neq A\subseteq \mathbb{R}$, $\min (\uparrow A)$ exists.
My book says that the candidate for $\min (\uparrow A)= \bigcup A$. That would mean that (for example) $\bigcup[0,1) = 1$ but why is it the case?
When I see finite unions of reals I can see the union is the maximum of the system, for example $\bigcup\{1,2,3\} = 3$ because of Dedekind's cut definition. So I can se that $\cup A \in A$. But what happen in infinite case (infinite refering to the cardinal of the set system in which I'm applying union) that it's not the case? ($\bigcup[0,1)$ as I mentioned above) Why $1=\cup [0,1] \in [0,1]$ even if $[0,1]$ is also infinite? Thanks in advance