TITLE: Importance of Least Upper Bound Property of $\mathbb{R}$
QUESTION [16 upvotes]: My professor asserts that the Least Upper Bound Property of $\mathbb{R}$ (Completeness Axiom) is the most essential piece in the study of real analysis. He says that almost every theorem in calculus/analysis relies directly upon on this Property.
I know that the Archimedian property of $\mathbb{R}$ directly uses the property for the proof, but I'm trying to think of other major theorems that use the property directly. Do you think he means consequences of the property? Because then the gates are wide open...
Does anyone know of any other theorems that use the properly directly?
Sorry this is more of a general question

REPLY [6 votes]: The importance is not so much because of the fact that the least upper bound property itself can be used directly to prove major thoerems.  As Gerry pointed out, the important point is that it says that the field of real numbers is complete, which is how it is distinguished from the field of rational numbers, for example.  Consequences of completeness, however it is formulated, are what is behind all of the analysis that would be impossible with $\mathbb Q$.  Compactness of closed and bounded intervals, connectedness of intervals, convergence of bounded monotone sequences, convergence of Cauchy sequences, nonemptiness of intersections of nested sequences of closed bounded intervals, and existence of limit points of bounded infinite subsets, are some of the important properties in analysis that $\mathbb R$ has and $\mathbb Q$ doesn't, and all are consequences of some formulation of completeness.  (Contrast this with the Archimedean property, which also holds for $\mathbb Q$.)
One could take other axioms as a starting point for analysis.  For example, completeness for metric spaces is formulated in terms of Cauchy sequences, mentioned in Arturo's and Thomas's answers, and that can also be done for $\mathbb R$.  There is one difference, however, which is that convergence of Cauchy sequences is not enough to guarantee the Archimedean property, so the latter would have to be taken as an additional axiom.  Arturo's answer shows that the intermediate value property for continuous functions on intervals could also be taken as the defining axiom.  I have seen at least one analysis textbook, Körner's excellent A companion to analysis, that makes convergence of bounded monotone sequences the defining axiom of the real numbers instead of the least upper bound property; the equivalence of the two is the topic of a recent question here.