Least Upper Bound Property

Every nonempty set of Real Numbers that is bounded above has a supremum (Least upper bound).

1. If $S\subset R$
2. If $S\ne \varnothing$
3. If $S$ is bounded above
4. Then $sup(E)$ exists

Rational Incompatibility

This is not a property of Rational Number. Every nonempty set of Rational Numbers ($S\subset Q$) has no supremum $Q$ Example: $S=\{x\in Q|x^2<2\}$ Then $sup(S)=\sqrt{2}$ which is Irrational