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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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  • $\begingroup$ 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"! $\endgroup$ – Bruno Stonek Dec 10 '11 at 22:08
  • $\begingroup$ @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 here $\endgroup$ – Arturo Magidin Dec 10 '11 at 22:46
  • $\begingroup$ @ArturoMagidin thank you for the advice. I'm somewhat new. $\endgroup$ – pigishpig Dec 10 '11 at 22:48
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One way to spot the importance of the Least Upper Bound property is to think about results that work in $\mathbb{R}$ but fail in $\mathbb{Q}$; this, because $\mathbb{Q}$ still has a lot of the algebraic properties of $\mathbb{R}$ (including the Archimedean property), but does not satisfy the Least Upper Bound property. So results that are true in $\mathbb{R}$ but are false in $\mathbb{Q}$ are good candidates for needing the Least Upper Bound property.

Some examples:

  1. The Intermediate Value Theorem uses the Least Upper Bound Property of $\mathbb{R}$ (in fact, the IVT is equivalent to the Least Upper Bound Property; see below).

  2. The fact that Cauchy sequences converge in $\mathbb{R}$ depends on the Least Upper Bound Property; without it, you can have sequences that are Cauchy but do not converge (as you do with $\mathbb{Q}$. That Cauchy sequences converge is very important in, for example, the definition of integration as limits of Riemann sums.

  3. Bounded sequences have convergent subsequences (the Bolzano-Weierstrass Theorem); the proof in $\mathbb{R}$ rests on showing that such a sequence has a monotone subsequence, and the proof that monotone bounded sequences converge uses the Least Upper Bound Property directly.


The equivalence of the IVT and the Least Upper Bound Property:

IVT. Let $f\colon [a,b]\to\mathbb{R}$ be a continuous function. If $f(a)\lt 0$ and $f(b)\gt 0$, then there exists $c\in(a,b)$ such that $f(c)=0$.

Least Upper Bound Property. Let $S\subseteq \mathbb{R}$ be a nonempty set that is bounded above. Then $S$ has a least upper bound bound in $\mathbb{R}$.

Theorem. IVT is equivalent to the Least Upper Bound Property.

Proof. Assume the Least Upper Bound Property, and let $f\colon [a,b]\to\mathbb{R}$ be continuous. let $S=\{x\in [a,b]\mid f(x)\lt 0\}$. Then $S$ is bounded above by $b$, and is nonempty, since $a\in S$. Let $c$ be the least upper bound for $S$. Note that $a\lt c\lt b$.

If $f(c)\neq 0$, then there exists $\delta\gt 0$ such that for all $x$, $|x-c|\lt\delta$, $|f(x)-f(c)|\lt \frac{|f(c)|}{2}$. In particular, $f(x)$ and $f(c)$ have the same sign.

But: if $f(c)\gt 0$, then this means that for all $\delta\gt 0$ there exists $s\in S$ such that $c-\delta \lt s\leq c$, so $f(s)\lt 0$, hence $|f(s)-f(c)| = f(c)+|f(s)|\gt \frac{f(c)}{2}$, a contradiction. In particular, $c\lt b$.

And if $f(c)\lt 0$, then there exists $x\in [a,b]$ with $x\lt c+\delta$ (since $c$ cannot be $b$), and then $f(x)\lt 0$, so $x\in S$, contradicting the fact that $c$ is an upper bound for $S$.

Therefore, we cannot have $f(c)\neq 0$, so $f(c)=0$, with $a\lt c\lt b$, and we are done.

(The following is an argument suggested to me some years ago by Lee Rudolph on sci.math) Conversely, assume that we have the Intermediate Value Theorem. Let $S$ be a nonempty subset of $\mathbb{R}$ that is bounded above; let $Y$ be the set of upper bounds of $S$, and define $f$ as follows: $$f(x) = \left\{\begin{array}{ll} 1 & x\in Y\\ -1 & x\notin Y \end{array}\right.$$ Note that there are points where $f$ takes value $-1$, and points where $f$ takes value $1$, but there are no point where $f$ takes the value $0$. That means, by the Intermediate Value Theorem, that $f$ is not continuous. Therefore, there is a point $z$ where $f$ is not continuous.

If there exists $s\in S$ such that $z\lt s$, then letting $\delta=s-z\gt 0$ we have that $(z-\delta,z+\delta)\cap Y = \emptyset$, and therefore $f$ is constant on this interval, hence continuous. So no such $s$ can exist, and therefore $z$ is an upper bound for $S$.

If there exists $y\in Y$ such that $y\lt z$, then letting $\delta=z-y\gt 0$ we have that $(z-\delta,z+\delta)\subseteq Y$, and so $f$ is constant on this interval, hence continuous. This contradiction tells us that no such $y$ can exist either.

So $z$ is in $Y$ and is a lower bound for $Y$, hence $z$ is the least element of $Y$. Thus, $z$ is the least upper bound of $S$, and $S$ has a least upper bound, as desired. QED

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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.

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I think that just the fact that we define almost everything in analysis in terms of limits in enough to see that this is fundamental ; since one way to define limits is to say that the inferior limit is equal to the superior limit, we need those somehow implicitly to speak of limits, and thus need the completeness axiom.

But more explicitly, you can think about the theorem that states that monotone increasing bounded above sequences converges : what you do is consider $$ \alpha = \sup_{n \in \mathbb N} \{ x_n \} $$ which exists by the completeness axiom. Then you use $\delta$'s and $\varepsilon$'s to show that $\alpha$ is actually the limit.

Another example would be the definition of $n^{\text{th}}$ roots in analysis, which are defined in terms of supremums. Basically everything you know that is defined in terms of supremums uses that axiom implicity.

Hope that helps,

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Just imagine trying to do analysis/calculus if all you had was the rational numbers, and you'll see how important completeness is.

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The fact that every bounded non-decreasing sequence has a limit is a direct result of this theorem.

The fact that every bounded sequence has a convergent subsequence is a relatively direct consequence.

From there, you can get that any Cauchy sequence is convergent.

There are two essential ways to define the reals from the rationals.

The first is to define the reals as Cauchy sequences of rational numbers upto some equivalence relation. This definition assures directly the completeness of the reals.

The second uses something called Dedekind cuts on the rationals. This technique assures directly the Least Upper Bound property of the reals.

Both definitions are equivalent, so it is a matter of emotion to say which property is more primary. I suppose that the least upper bound property is easier to state in predicate logic as an axiom of the reals, so, from a logician's point of view, it might be preferable.

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