I'm trying to work out a question of a past exam, but I can't seem to figure it out. If I could get some help, I would be very appreciative. The question is:
When we use the definition of real numbers as sets of rational numbers, the supremum of a set $A$ of numbers is defined as the number corresponding to the union of the corresponding sets. Specifically, if we write $\alpha_x$ for the set of rational numbers which corresponds to $x\in A$, then $\sup A$ is the number of corresponding to the set $\cup_{x\in A} \alpha_x$. Show that $\sup A$ defined in this way is indeed the least upper bound of the numbers in $A$.
Thanks in advance.