Conversation 8: Defining the real numbers
Winfried Just
Department of Mathematics, Ohio University
Companion to Advanced Calculus
Does every Cauchy sequence of real numbers converge?
Cindy: So is it true then that every Cauchy sequence of real numbers converges? I’m curious.
Alice: We will be able to figure this out.
But first we need a formal definition of the real numbers.
Denny: Didn’t our prof already kinda say how they are defined?
As limits of convergent sequences of rational numbers, as I recall.
Bob: But for any given real number x there are many different sequences of rational approximations that converge to x , as we have seen for x =√
2.
Theo: Therefore the textbook essentially defines the real numbers as certain equivalence classes of the relation ∼ on sequences of rational numbers.
Question C8.1: Why do we consider only the restriction of the equivalence relation ∼ to sequences of rational numbers in this definition, and not the relation ∼ on all sequences of real numbers?
We cannot use all sequences in this definition
Theo: We cannot use sequences of real numbers to define real numbers. This would lead to circularity. Therefore the definition is based on certain sequences of rational numbers, which were already defined previously.
Bob: But we cannot use a sequence like (an)∞n=0= ((−1)n)∞n=0 either, as it goes nowhere.
Theo: Right! I said “certain sequences of rational numbers.”
Denny: Your “certain” then means “convergent”, Theo.
Why did you not say this in plain language?
Alice: Good question, Denny!
Let us recall what “convergent” means.
Bob: A sequence (an)∞n=m is convergent iff there exists a real number x such that limn→∞an= x .
Question C8.2: Why did Theo not say “convergent”?
We can use only Cauchy sequences in this definition
Theo: Had I said “convergent,” I would in effect have said “there exists a real number with a certain property.”
But I cannot make such a statement in the definition of real numbers without creating circularity.
However, we can define the real numbers as equivalence classes of Cauchy sequencesof rational numbers. This does not require prior knowledge of real numbers and does not create circularity.
Cindy: Is this because (an)∞n=m is Cauchy if, and only if
∀ε > 0 ∃N ≥ m ∀j, k ≥ N |aj − ak| ≤ ε,
so that we need only inequalities and absolute values for rational numbers, which were already defined?
Theo: Right.
Frank: But there is still a real number ε in this definition.
A definition of real numbers
Theo: Granted. Throughout Lecture 20, it was assumed that the real numbers were already defined, so that, in particular, the ε in the definition of a Cauchy sequence could be any positive real number. But we can define Cauchy sequences by considering only rational ε, so that Cindy’s formula becomes:
∀ε ∈ Q (ε > 0 =⇒ ∃N ≥ m ∀j, k ≥ N |aj − ak| ≤ ε).
This is what the textbook does. We will see in Module 21 that it does not matter whether we allow only rational values for ε or real values. However, we can see here that this restriction eliminates all circularity from the definition.
Then the textbook defines real numbers x as equivalence classes of Cauchy sequences under the equivalence relation ∼:
x := [(an)∞n=m]∼
and also defines
limn→∞an:= [(an)∞n=m]∼.
The notation of the textbook
Bob: Your definition doesn’t look at all like the textbook definition, Theo!
Cindy: But it makes more sense to me than the notation and definitions in Sections 5.1 through 5.3 of the textbook, which are so confusing.
Alice: It seems the textbook author was trying to avoid formally defining equivalence relations and equivalence classes. This then forced him to use this confusing notation. Our instructor thought it better to have us first learn the general concept of an
equivalence class. This material was not easy, but once we have understood it, the constructions of all the number systems become a lot more transparent.
Theo: As in the definitions of the integers and the rationals, our notation differs from that of the textbook. But for the reasons Alice mentioned, our definition is essentially the same as the one in the book, only written in a better terminology.
Treating rational numbers as real numbers
Alice: We can now identify each rational number q with the equivalence class [(q)∞n=0]∼ of a constant, and hence Cauchy, sequence that has all terms equal to q.
In this way the set Q of rational numbers can be treated as a subset of the set R of real numbers.
Theo: Intuitively yes, although formally these are different types of mathematical objects.
Frank: Come on, Theo! If you want to be that formal, you should have defined your limits not
as limn→∞an:= [(an)∞n=m]∼, but as limn→∞[(an)∞k=0]∼:= [(an)∞n=m]∼.
Denny: I think you meant limn→∞[(an)∞n=0]∼:= [(an)∞n=m]∼. Question C8.3: Who is right, Frank or Denny?
Cauchy sequences of rational numbers are convergent
Frank: Nope! In my notation, (an)∞k=0 is a constant sequence for each fixed n. This constant sequence would then be interpreted as the rational number an, as Alice had told us. This is why I used a new letter k. And it gives us the equality:
limn→∞[(an)∞k=0]∼= limn→∞an.
Denny: OK, OK. But I like Theo’s original notation better.
Cindy: So would this then mean that every Cauchy sequence of rationalnumbers converges to some real number?
Theo: Right. This is trueby definition,since we have specified a limit for each such sequence.
Cindy: But this does not quite answer my question whether every Cauchy sequence ofrealnumbers is convergent.
Theo: Right. And we cannot answer it yet.
Denny: Why not?
What do we need to talk about convergence of sequences of real numbers?
Theo: In order to rigorously talk about convergence of real numbers, we would need to define the notion of ε-closeness of two real numbers. In our formal notation, this would mean
|[(an)∞n=m]∼− [(bn)∞n=m0]∼| ≤ [(εn)∞n=m00]∼
Denny: Looks horrible to me.
Can we somehow get away without using this notation?
Theo: Yes. In fact, we will use the so-calledupper bound property of the ordering of the reals for deriving a full answer to Cindy’s question without using this complicated notation.
Bob: But either way, would we not need to define differences of real numbers, their absolute values, and their ordering first before we can formally define ε-closeness of two reals x and y , even if we express the concept in the familiar notation |x − y | ≤ ε?
Theo: This is correct.
Defining arithmetic operations on the reals
Cindy: Wait! Based on what we learned in Modules 19 and 20, for any Cauchy sequences (an)∞n=m and (bn)∞n=m we can simply define:
[(an)∞n=m]∼+ [(bn)∞n=m]∼:= [(an+ bn)∞n=m]∼,
Denny: [(an)∞n=m]∼− [(bn)∞n=m]∼ := [(an− bn)∞n=m]∼, Bob: [(an)∞n=m]∼× [(bn)∞n=m]∼:= [(anbn)∞n=m]∼, Frank: and [(a[(bn)∞n=m]∼
n)∞n=m]∼ := [(an/bn)∞n=m]∼.
Question C8.4: Did Cindy, Denny, Bob, and Frank get this right?
Alice: Yes, we can define addition, subtraction, and multiplication of real numbers in this way. But for division we need to make the additional assumption that the sequence (bn)∞n=m is bounded away from 0 so that it does not represent the real number 0.
It then follows from Proposition M19.2 that these operations are well-defined, and from Proposition M20.1 that the results of these operations are equivalence classes of Cauchy sequences, that is, real numbers.
(R, +, ×, 0, 1) is an algebraic field
Theo: The real numbers with these operations form an algebraic field. That is, they have all the properties that were listed on slide 7 of Lecture 14 for the rational numbers.
Cindy: Is this because, like for the distributivity law x (y + z) = xy + xz,
we would have:
[(an)∞n=m]∼× ([(bn)∞n=m]∼+ [(cn)∞n=m]∼] = [(an(bn+ cn)∞n=m]∼, by our definitions,
and [(an(bn+ cn)∞n=m]∼= [(anbn+ ancn)∞n=m]∼
by the same distributivity law for the rationals, and
[(anbn+ancn)∞n=m]∼= [(an)∞n=m]∼×[(bn)∞n=m]∼+[(an)∞n=m]∼×[(cn)∞n=m]∼]
by our definitions again?
Theo: Exactly! The other properties of an algebraic field can be proved in a very similar way.
Take-home message
We define the real numbers as equivalence classes of Cauchy sequences of rational numbers.
We will somewhat informally identify each rational number q with equivalence class [(q)∞n=0]∼ of a constant, and hence Cauchy, sequence that has all terms equal to q. Then Q ⊆ R.
Any Cauchy sequence (an)∞n=m of rational numbers is convergent in the set of real numbers and its limit is defined as
limn→∞an:= [(an)∞n=m]∼.
The arithmetic operations on the real numbers can be defined in a natural way in terms of operations on the terms of suitable
representatives of the equivalence classes. Division by 0 is ruled out by requiring that the sequence that represents the denominator be bounded away from 0.
The structure (R, +, ×, 0, 1) is an algebraic field.
