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