The supremum axiom The axiom says: every nonnempty set bounded above has a supremum. In the case of $\\{ q \in \mathbb{Q} \mid q^2\le2\\}$ we do not have a supremum in $\mathbb{Q}$, but the one in $\mathbb{R}$. Is it necessary for the supremum to be in the same set? If so, why? The set I wrote here will be still bounded by $\sqrt{2}$, won't it?

In $\mathbb Q$, the set $\\{q\in\mathbb Q\,|\,q^2\leqslant2\\}$ has no supremum. In $\mathbb R$, it has one (which is $\sqrt2$).