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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

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    The conclusion of Bolzano's theorem (intermediate value theorem) would be false if $\mathbb{R}$ weren't complete. Now think of all the times you say "since $f$ is a continuous function, negative here and then positive, it must be zero somewhere in the middle"!2011-12-10
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    @pigishpig: You may want to hold off before accepting an answer for several reasons: (i) You are more likely to attract answers if the question does not have an accepted answer than if it does; I think the particular kind of question you are asking is also one that can attract a lot of good answers that don't overlap. (ii) It's always a good idea to let a question lie for at least a couple of hours before accepting an answer, unless it's something that has a definitive, clear, singular answer. Note you can only accept one question, so you may want to think a bit on which one you'll accept here2011-12-10
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    @ArturoMagidin thank you for the advice. I'm somewhat new.2011-12-10

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