For a problem set, I have to show that the set $\mathbb{Q}_{x} :=$ set of all rational numbers for which $q \leq x$ has a supremum in $x$.
My attempt is to suppose that there is a $y < x $ that is an upper bound of $\mathbb{Q}_{x}$ and then find a $q \in \mathbb{Q}_{x} > y$. Unfortunately, I am not allowed to use the fact in between any arbitrary real numbers lies a rational one (as this is part of a proof to show that the rational numbers are dense in the real ones...)
I would appreciate any hints greatly, as I find myself banging my head against a wall with this simple problem for some time now...