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…
Best Answer
Consider $x - y$ and use the Archimedean property. Of course, I'm assuming that you've already built up $\mathbb{R}$ axiomatically, which might not be the case. If you haven't already developed the real numbers in this way, you should probably include more information about what has been covered and what tools are at your disposal.