Class 2 – Notes (1-13-16)
Math 3114 – Higher Analysis
Winter Term 2015-16
Daily Syllabus – Part I (tentative)
Class
1
Date
M-1/11
2
W-1/13
3
4
5
6
7
8
9
10
11
F-1/15
M-1/18
W-1/20
F-1/22
M-1/25
W-1/27
F-1/29
M-2/1
W-2/3
Topic
Introductory remarks; Discussion of Section 1.1 – The irrationality of 2; Discussion
of Section 1.2 – Some Preliminaries
Discussion of exercises from Section 1.2; Discussion of Section 1.3 – The Axiom of
Completeness
TBA
MLK Holiday – no class
TBA
TBA
TBA
TBA
TBA
TBA
Exam 1
Quiz and homework discussion
Summary of Section 1.3 - Axiom of Completeness
This section of the text (and the next) explores two related questions –
 What are the real numbers?
 What makes the real numbers special?
As indicated previously, the actual construction of the real numbers (from first principles) is postponed to the end of the
text.
In essence the set of real numbers are special because one can do both arithmetic and "topology" (measure distances
between numbers, hence have a notion of "closeness") and the two concepts are "compatible".

The set of real numbers () is a field and naturally includes , , and , namely       
o
To be a field, briefly means that there are two arithmetic operations (+, ) defined on , the operations satisfy the
standard properties of arithmetic, and one can divide by nonzero members of .
o
There are many examples of fields besides  – e.g., the rational numbers , the complex numbers , and the
modular fields p where p is a prime number. On the other hand some important sets of numbers are not fields –
e.g., the set of natural numbers , the set of integers , and the modular rings m where m is a composite integer.
o
Recall that the complex numbers  is a field that naturally contains  and has the important algebraic property
that the Fundamental Theorem of Algebra holds for it. (This is not true for , a major deficiency of .)

The set of real numbers is an ordered field.
o Briefly this means there is an order relation (<) defined on , this order is a total order (i.e., every two nonequal numbers can be compared, hence we have the trichotomy law: given a and b, either a<b, a=b, or b<a) and this
order relation is compatible with the arithmetic operations + and , namely if a < b then a + c < b + c and if a < b
and 0 < c then ac < bc. In particular it follows that if a  0 then a2 > 0.
o
That  is an ordered field, allows one to defined the concepts of "bigger than zero" (positive elements, and hence
negative elements), absolute value, and hence distance (namely, dist(a,b) = | a – b |). Actually, having such an
order and having a set of positive elements is an equivalent way to define an ordered field.
There are other ordered fields, e.g. .
o
 is not an ordered field (why? – look at i2).
o
Remark 1. Using language like "the set of real numbers is a field" is a little "sloppy". We should really have said: the set of
real numbers together with the operations of + and  is a field and the set of real numbers with the operations of + and 
and the relation < is an ordered field. Thus by  we will mean both the "set" and the field and the ordered field – hopefully
in context it will be clear what we mean.
Remark 2. As noted above, the concept of absolute value allows one to introduce the concept of distance, which is critical
for notions like limit, continuity, and convergence. If one looks for ways to extend these notions to others sets. e.g., 2 or
3 there are several ways . One way is to introduce the concept of "metric space" – a place where one has the concept of
"distance" (or even more generally, a topological space). Another way that is useful in a vector space (where one has
addition and scalar multiplication) is to introduce the concept of "normed space" ( a generalization of a space with an
absolute value).


The set of real numbers is an archimedian ordered field.
o Briefly this means for every x  , there exists n   so that n > x. ( has no "infinite elements".)
o
Corollary: For every positive   , there exists n   so that
o
o
There are other archimedian ordered fields – e.g., .
The text discusses this property in section 1.4
1
  . ( has no "infinitesimal elements".)
n
The set of real numbers is a complete ordered field. (!) In order to state this property we must introduce some
important concepts. Let A  .
o
o
o
o
o
o
o
o
Definition: We say A is bounded above if there exists a real number b so that a  b for all a A. We call such a
b an upper bound for A.
Definition: An element b   is said to be a least upper bound for A if
(i) b is an upper bound for A, and
(ii) if c is any upper bound for A, then b  c.
Examples (?): sets that have upper bounds and sets that do not have upper bounds
Examples (?) : upper bounds and least upper bounds of a set need not belong to the set
An important (but easy) observation: If a least upper bound for a set exists, it is unique. (Proof?) Thus if A has a
least upper bound , it is legitimate to say the least upper bound for A, and in such a case we will denote it by
lub(A).
Alternative terminology. A least upper bound is often called a supremum for the set, and is denoted by sup
rather than lub.
There are analogous concepts: bounded below; greatest lower bound (glb), infimum (inf). (These concepts will
be developed as exercises.) We say a set is bounded if it has both an upper bound and a lower bound.
There is a concept loosely related to least upper bound (respectively greatest lower bound), namely maximum
(resp. minimum) of a set: An element M (resp. m) is called a maximum (respectively minimum) of a set A if M
 A (respectively m  A) and a  M (resp. m  a) for all a  A. Observe that a set could have a least upper
bound but not a maximum and similarly for infimum and minimum. (example?).
Remark 3. It is often convenient to extend the real numbers to include - and + called the extended real number system.
One defines a natural arithmetic and order that includes these symbols. In this context, one can extend "sup" and "inf" to all
sets. For example, is A is nonempty and not bounded above then sup(A) = + and if A is the empty set then sup(A) = -.

 is a complete ordered field – this means, every nonempty set of real numbers that is bounded above has a least
upper bound. (Is this true if we do not assume "nonempty"?)
o The text chooses to take this as an axiom – i.e., we assume its truth without proof. If one starts off more basic, i.e.,
gives a construction of  from  (see chapter 8) then one would need to prove this property as a theorem.
o
The set of rational numbers  is not complete. (Why?) The text shows in section 1.4 that the fact that  is
o
archimedian follows from the fact that  is complete so we see archimedian is a weaker notion than complete.
The corresponding statement for greatest lower bound follows from the statement about least upper bounds as a
theorem (Can you give a proof?)
o
It is of theoretical interest that if a set is abstractly an archimedian ordered field then it is "isomorphic" to a
subfield of . Moreover, if a set is a complete ordered field then it is isomorphic to . Thus  is completely and
uniquely characterized by these properties. Note this also implies the set of complex numbers  is not a complete
ordered field. It would be of interest to explore what goes wrong and what can be salvaged (because one would
like to extend the calculus to ).

How does one show for a given set A that an element b is a least upper bound for A?
o Of course one way is to simply look to the definition – i.e., show that b is an upper bound for A and that it has the
property that b  c for every upper bound c.
o For convenience the text proves a useful lemma that is sometimes easier to use.
 Lemma. Let A be a set of real numbers and let b be an upper bound for A. Then b = sup(A)  for every  >
0, there exists an element a  A so that b -  < a. (Proof of Lemma --see text or class discussion)

Summary. The properties of  and real-valued functions of a real variable will now be developed assuming  is a
complete ordered field.
Assignment
 Read carefully and thoroughly Section 1.3 of the text
 HW #2 (due Friday, Jan 15)
o A: p. 11: 9, 10, 11, 13
o B: p. 11: 8, 12