Completeness Axiom and its significance

The general statement is: Any subset of real numbers that is bounded above has a supremum. Now this property is not shared by other number systems.

We need the following to do calculus and we ask the question what makes real numbers so special over rational numbers
It should be a complete ordered field. Only real numbers are complete ordered field.
1. We want our system to be an Abelian Group under Addition.
2. We also want it to be a field.
3. We need total Ordering

Both and satisfies all the above three properties and Integers satisfy the first and third. The missing link is the completeness axiom.
Upper bound