Axiom 34: The Completeness Axiom

If a set $A\subset \mathbb{R}$ has an upper bound, then the supremum of $A$ exists.

Similarly, if a set $A\subset \mathbb{R}$ has a lower bound, then the infimum of $A$ exists.

Why is this axiom important?

As we'll see, this is the axiom that distinguishes the real numbers from the rational numbers; it means that every sequence that eventually "settles down" converges to some real number.