When doesn't a supremum exist? Other than ∞, is there another case where a supremum (or an infimum for that matter) doesn't exist?

> **Supremum axiom** : Any nonempty subset $A\subset\mathbb{R}$ which is bounded above has a supremum $s\in\mathbb{R}$.

This is only valid in $\mathbb{R}$. For example, the set $\\{x\in\mathbb{Q}:x\geq 0, \ x^{2}<2\\}$ does not admit supremum, since $\sqrt{2}\
otin\mathbb{Q}$.