Notes on Real Analysis, 3A03
Prof. S. Alama
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Revised January 20, 2020
Introduction
Real Analysis is the study of the real numbers
R
and of real valued functions of one or more
real variables.
If this sounds familiar, it should: you’ve been doing calculations with real
numbers and functions for several years now, starting in high school and continuing through
calculus.
On the other hand, most of what you’ve done with real numbers and functions
has involved applying “rules” intended to justify which kinds of calculations are appropriate
(in the sense that they lead to correct results,) which might have seemed mysterious and
arbitrary to you at the time.
The goal of this course is to develop some understanding
of functions of a real variable using logical deductive reasoning, and in doing so obtain a
much firmer grasp of how these “rules” may be justified and how to determine whether any
computational procedure will always yield correct results.
One of the great mysteries, in fact, is to figure out what the real numbers actually
are.
You’ve learned that they include all of the other more familiar number systems from elemen
tary mathematics: the counting (natural) numbers
N
, integers
Z
, rational numbers (fractions
of integers)
Q
, and algebraic numbers (ie, roots of polynomials with integer coe
ffi
cients, such
as
p
2
,
3
p
5
, . . .
.) In high school you mostly learned the
algebra
of real numbers. Amazingly,
one cannot really understand the reals via algebra.
Their fundamental properties involve
analysis– basically, one must understand some sense of
limits
.
Let’s summarize some algebraic and geometric properties of the reals, which you aleady
know but perhaps haven’t thought of in these terms:
•
R
is a
field
:
If
a, b, c
2
R
, then so are
a
+
b
,
ab
, and
b/c
if
c
6
= 0.
Addition and
multiplication are commutative,
a
+
b
=
b
+
a
and
ab
=
ba
, associative,
a
+ (
b
+
c
) =
(
a
+
b
) +
c
and
a
(
bc
) = (
ab
)
c
, and distributive,
a
(
b
+
c
) =
ab
+
ac
. Each
a
2
R
has an
additive inverse

a
2
R
, and a multiplicative inverse 1
/a
2
R
, provided
a
6
= 0.
•
R
is an
ordered
set. If
a
6
=
b
are distinct real numbers, then either
a < b
or
b < a
.
In addition, the ordering is transitive (ie, if
a > b
and
b > c
, then
a > c
,) and is
consistent with the algebraic operations. In particular, if
a, b, c
2
R
with
a < b
then
a
+
c < b
+
c
. If
c >
0, then
ac < bc
, but if
c <
0, then
ac > bc
.
1
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1
•
R
has a “metric property”, determined by the absolute value,

x

=
(
x,
if
x
≥
0,

x,
if
x <
0.
Then, the
distance
between
a, b
2
R
is defined by
d
(
a, b
) =

a

b

, the length of the
line segment joining the two points on the number line.
We say
R
is an
ordered field.
However, the rational number set
Q
is also an ordered
field! The distinction between
R
and
Q
, and the intimate relationship between them form
an important theme in the course.
To see how the algebraic and ordering properties are
defined, and how the various familiar “rules” are justified from the axioms, see Section 2 in
the Bartle & Sherbert textbook.