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Does it simply depend on if we are in $\mathbb{R}$ or $\mathbb{Q}$, and whether the supremum exists in this set? like a set of all $a^2 < 2$ will be upper-bounded in $\mathbb{Q}$, but won't have a supremum, and in $\mathbb{R}$ it will be both upper-bounded and have a supremum?

Does it always work like that and is this the only difference?

Lucky Chouhan
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Nila Alesich
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    Related: see this – Lucky Chouhan Sep 22 '23 at 16:58
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    Let $A\subset \mathbb{R}$. If $A$ is bounded then a supremum of $A$ exists and it is a real number $s:=\sup A$ (by the least upper bound property/axiom of the reals). The supremum $s$ may or may not be in $A$. If we replace $\mathbb{R}$ with the rationals, then $A$ can be bounded, but no least upper bound necessarily exists, as you pointed out with a standard example. – Nap D. Lover Sep 22 '23 at 17:14

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