Axiom of completeness

The Axiom of completeness is a foundational idea in mathematics and real analysis that ensures the real number line has no gaps. It states that every non-empty set of real numbers that has an upper limit (a boundary that numbers don't surpass) also has a least upper limit, called the supremum. In simple terms, if a set of numbers is bounded above, there is always a smallest number that is still greater than or equal to every number in that set. This property guarantees the real number line is continuous and complete, allowing for consistent analysis and calculations.