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I'm now going through the completeness axiom. What is it used for? What can you do with it? Why is it called completeness? And beside that, how can you proof this theorem? Suppose that S is a nonempty subset of R and k is an upper bound of S. Then k is the least upper bound of S if and only if for each $\epsilon > 0$ there exists $s \in S$ such that $k - \epsilon < s$. I tried picking a random $\epsilon$, but then I come to the point $k-s<-\epsilon$..

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    You refer to it first as an axiom, then as a theorem. It can't be both, so which one is it?If it is an axiom, it is assumed , and needs no proof.2011-10-26
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    And you made a mistake in going from $k-e< s$ to $k-s< e$.2011-10-26
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    @gary: The axiom says that every bounded subset has a least upper bound. The second is a theorem to characterize this least upper bound.2011-10-26
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    The theorem was an exercise where you might use the completeness axiom. But I couldn't find the answer on it.2011-10-26
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    Verb: Prove (the thing you *do*). Noun: Proof (an often logical derivation describing why a theorem holds).2011-10-26
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    @Asaf: so which is the second one, i.e., what is the statement of the theorem, is it that every Cauchy sequence of reals converges?2011-10-26
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    @gary: The theorem is that if $k$ is *an* upper bound of $S$, then $k$ is the least upper bound if and only if for every $\epsilon>0$ there is some $s\in S$ such that $k-\epsilon$$\begin{align}{}\\\end{align}$$ The axiom of completeness is that every bounded subset of $\mathbb R$ has a least upper bound. – 2011-10-26
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    @Asaf:Got it, thanks.2011-10-26
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    @Kevin: Just a note on how you phrased things. Picking a "random $\epsilon$" is not what what you mean to say. For instance, considering $0 < \epsilon$ we could pick a "random $\epsilon$" uncountably many times and find that it is true every single time. That doesn't make the statement $0 < \epsilon$ true for all $\epsilon$. A better phrasing is "I let an $\epsilon$ be given".2011-10-26
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    The completeness axiom formalizes the idea that any number you think should be there is actually there. The "existence of the $\sup$" is a precise and handy form of this principle.2011-10-26

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The completeness axiom is probably the most important concept in real analysis. Every theorem in real analysis follows from it; for instance, every convergent sequence of real numbers has a real limit (which is not the case for, say, rational numbers, which are not a complete field). The fact that real numbers are a continuum (which is implied by completeness) allows you to derive most results in calculus, etc.

Also, as its name implies, the completeness axiom is an axiom, not a theorem, therefore, there's no proof for it (at least if you're using an axiomatic, that is, non-constructive, definition of real numbers).

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    Is there a reason these are two separate answers?2011-10-26
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    Not really, just the fact that the OP asked two different questions. I can merge them if it seems appropriate.2011-10-26
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    No, I guess it's OK. I usually only write separate answers if I have two disjoint approaches to the question. If the question consists of disjoint parts, I usually ask the OP to post two separate questions, but I guess that might have been overkill in this case.2011-10-26
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"Completeness" refers to a wide variety of results that describe how the real line is, well, 'complete' -- that there are no holes, that it's not missing anything that it ought to have.

For example, consider the "Elementary Continuity Principle" from Euclidean geometry:

Let C be a circle. Let P be a point inside of C. Let Q be a point outside of C. Then, the line segment PQ intersects C.

This type of geometric fact is an example of something that depends on completeness. An algebraic example (which is effectively the same as the above) is

Every positive real number has a square root.

With just rational numbers, for example, both of the above fail. For example, 2 doesn't have a square root -- the rational numbers are "missing" some numbers. A counterexample to the geometric version is to consider the rational plane (every point has rational coordinates), and let

  • C be the circle centered at (0,0) with radius 1
  • P be the origin (0,0)
  • Q be the point (1,1)

Then the line segment PQ starts inside the circle, ends outside the circle, but doesn't ever intersect the circle -- the circle has "holes"!

Whatever particular statement you have been introduced to as the "completeness axiom" was chosen for the sake of pedagogy and/or tradition. You shouldn't get the impression that that axiom is what completeness 'means' -- instead, that axiom is just one of a wide variety of facts related to completeness.

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As for the theorem you're trying to prove: Suppose $k$ is the least upper bound of $S$ and that there's an $\epsilon$ such that $k-\epsilon>s$ for any $s\in S$; but then $k-\epsilon$ is an upper bound of $S$ and $k-\epsilon

This fact can be used to prove that, given a bounded, non-empty set $A$, there's a sequence $(x_n)$ of elements of $A$ that converge to $a$, where $a$ is $A$'s least upper bound. You can construct such a sequence choosing $x_n\in A$ such that $a−\frac 1n0$, $|x_n−a|<\epsilon$ for a big enough $n$. This uses the fact we have proven to choose an appropriate $x_n$. As a sidenote, we can choose the smallest $x_n$ possible at each step to avoid using the axiom of choice.

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    Yes, thanks! That makes sense. Can you tell me where this is used for?2011-10-26
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    Hmm, this is just a nice fact about least upper bounds. It doesn't imply anything of importance by itself but it's a nice little tool that appears in several proofs of more important theorems.2011-10-26
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    Interesting, what kind of proofs? We now have a lot of those little theorems, but I can't see why they are important at this point.. So I'm curious.2011-10-26
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    Thanks :-)! How can I mark this as 'answer' ;-)?2011-10-26
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    I added my comments as part of my previous answer.2011-10-26
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    Regarding your avoidance of (AC): How do you know that "the smallest possible x_n" lies in A? It seems that you are assuming that A is closed and then taking the infimum of the intersection of A with each of these intervals. What if A is not closed?2015-03-26
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completeness axiom is ''every non empty subset of R which is bounded above has the least upper bound'' or, ''every non empty subset of R which is bounded below has the greatest lower bound'' The above two are axioms so there is no need to prove it, if any one wants to prove 1st of axioms then he has to use 2nd or for trying to prove 2nd then must aply 1st.