Axiom for real numbers

Completeness Axiom of Real Numbers (Supremum Property)

Let $A$ be a non-empty subset of real numbers $\mathbb{R}$. If there exists an upper bound $n\in \mathbb{R}$ such that

$$a\le n,\forall a\in A,$$

then there exists a least upper bound (supremum) (x \in \mathbb{R}) such that

$$a\le x,\forall a\in A,$$

and for any $y<x$, there exists some $a\in A$ with $y<a$.

This axiom guarantees that every non-empty, bounded-above set of real numbers has a least upper bound in $\mathbb{R}$. It is a fundamental property that distinguishes the real numbers from the rational numbers and underlies limits, continuity, and integration in real analysis.