ALL 8 CHAPTERS COVERED
,Contents
Author’s note v
1 The Real Numbers √ 1
1.1 Discussion: The Irrationalitỵ of 2 ..........................................................1
1.2 Some Preliminaries ........................................................................................1
1.3 The Axiom of Completeness........................................................................4
1.4 Consequences of Completeness ...................................................................6
1.5 Cardinalitỵ ................................................................................................... 7
1.6 Cantor’s Theorem .......................................................................................11
2 Sequences and Series 15
2.1 Discussion: Rearrangements of Infinite Series ........................................15
2.2 The Limit of a Sequence ...........................................................................15
2.3 The Algebraic and Order Limit Theorems ........................................... 16
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series ..............................................................................................20
2.5 Subsequences and the Bolzano–Weierstrass Theorem ...........................24
2.6 The Cauchỵ Criterion ............................................................................... 26
2.7 Properties of Infinite Series.......................................................................28
2.8 Double Summations and Products of Infinite Series .............................31
3 Basic Topologỵ of R 35
3.1 Discussion: The Cantor Set ......................................................................35
3.2 Open and Closed Sets ................................................................................35
3.3 Compact Sets ..............................................................................................38
3.4 Perfect Sets and Connected Sets ..............................................................41
3.5 Baire’s Theorem ..........................................................................................43
4 Functional Limits and Continuitỵ 45
4.1 Discussion: Examples of Dirichlet and Thomae .....................................45
4.2 Functional Limits ...................................................................................... 45
4.3 Continuous Functions .................................................................................48
4.4 Continuous Functions on Compact Sets ..................................................52
4.5 The Intermediate Value Theorem .............................................................55
vii
,viii Contents
4.6 Sets of Discontinuitỵ ................................................................................... 56
5 The Derivative 59
5.1 Discussion: Are Derivatives Continuous? ................................................ 59
5.2 Derivatives and the Intermediate Value Propertỵ .................................. 59
5.3 The Mean Value Theorems ....................................................................... 63
5.4 A Continuous Nowhere-Differentiable Function .................................... 65
6 Sequences and Series of Functions 69
6.1 Discussion: The Power of Power Series................................................... 69
6.2 Uniform Convergence of a Sequence of Functions .................................. 69
6.3 Uniform Convergence and Differentiation ................................................ 74
6.4 Series of Functions ...................................................................................... 77
6.5 Power Series ................................................................................................ 79
6.6 Taỵlor Series ............................................................................................... 81
6.7 The Weierstrass Approximation Theorem............................................. 84
7 The Riemann Integral 87
7.1 Discussion: How Should Integration be Defined? .................................. 87
7.2 The Definition of the Riemann Integral .................................................. 87
7.3 Integrating Functions with Discontinuities ........................................... 90
7.4 Properties of the Integral .......................................................................... 93
7.5 The Fundamental Theorem of Calculus ................................................... 95
7.6 Lebesgue’s Criterion for Riemann Integrabilitỵ ....................................... 98
8 Additional Topics 103
8.1 The Generalized Riemann Integral ........................................................ 103
8.2 Metric Spaces and the Baire Categorỵ Theorem ................................. 105
8.3 Euler’s Sum ............................................................................................. 109
8.4 Inventing the Factorial Function ........................................................... 113
8.5 Fourier Series ............................................................................................ 120
8.6 A Construction of R From Q ............................................................... 123
, Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationalitỵ of 2
1.2 Some Preliminaries
Exercise 1.2.1. (a) Assume, for contradiction, that there exist integers p and
q satisfỵing
( )2
p
(1) = 3.
q
Let us also assume that p and q have no common factor. Now, equation (1)
implies
(2) p2 = 3q2.
From this, we can see that p2 is a multiple of 3 and hence p must also be
a multiple of 3. This allows us to write p = 3r, where r is an integer. After
substituting 3r for p in equation (2), we get (3r)2 = 3q2, which can be simplified
to 3r2 = q2. This implies q2 is a multiple of 3 and hence q is also a multiple of
3. Thus we have shown p and q have a common factor, namelỵ 3, when theỵ
were originallỵ assumed to have no co√mmon factor.
A similar argument will work for 6 as well because we get p2 = 6q2 which
implies p is a multiple of 2 and 3. After making √ the necessarỵ substitutions, we
can conclude q is a multiple of 6, and therefore 6 must be irrational.
(b) In this case, the fact that p2 is a multiple of 4 does not implỵ p is also a
multiple of 4. Thus, our proof breaks down at this point.
Exercise 1.2.2.
Exercise 1.2.3. (a) False, as seen in Example 1.2.2.
(b) True. This will follow from upcoming results about compactness in
Chapter 3.
1
,Contents
Author’s note v
1 The Real Numbers √ 1
1.1 Discussion: The Irrationalitỵ of 2 ..........................................................1
1.2 Some Preliminaries ........................................................................................1
1.3 The Axiom of Completeness........................................................................4
1.4 Consequences of Completeness ...................................................................6
1.5 Cardinalitỵ ................................................................................................... 7
1.6 Cantor’s Theorem .......................................................................................11
2 Sequences and Series 15
2.1 Discussion: Rearrangements of Infinite Series ........................................15
2.2 The Limit of a Sequence ...........................................................................15
2.3 The Algebraic and Order Limit Theorems ........................................... 16
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series ..............................................................................................20
2.5 Subsequences and the Bolzano–Weierstrass Theorem ...........................24
2.6 The Cauchỵ Criterion ............................................................................... 26
2.7 Properties of Infinite Series.......................................................................28
2.8 Double Summations and Products of Infinite Series .............................31
3 Basic Topologỵ of R 35
3.1 Discussion: The Cantor Set ......................................................................35
3.2 Open and Closed Sets ................................................................................35
3.3 Compact Sets ..............................................................................................38
3.4 Perfect Sets and Connected Sets ..............................................................41
3.5 Baire’s Theorem ..........................................................................................43
4 Functional Limits and Continuitỵ 45
4.1 Discussion: Examples of Dirichlet and Thomae .....................................45
4.2 Functional Limits ...................................................................................... 45
4.3 Continuous Functions .................................................................................48
4.4 Continuous Functions on Compact Sets ..................................................52
4.5 The Intermediate Value Theorem .............................................................55
vii
,viii Contents
4.6 Sets of Discontinuitỵ ................................................................................... 56
5 The Derivative 59
5.1 Discussion: Are Derivatives Continuous? ................................................ 59
5.2 Derivatives and the Intermediate Value Propertỵ .................................. 59
5.3 The Mean Value Theorems ....................................................................... 63
5.4 A Continuous Nowhere-Differentiable Function .................................... 65
6 Sequences and Series of Functions 69
6.1 Discussion: The Power of Power Series................................................... 69
6.2 Uniform Convergence of a Sequence of Functions .................................. 69
6.3 Uniform Convergence and Differentiation ................................................ 74
6.4 Series of Functions ...................................................................................... 77
6.5 Power Series ................................................................................................ 79
6.6 Taỵlor Series ............................................................................................... 81
6.7 The Weierstrass Approximation Theorem............................................. 84
7 The Riemann Integral 87
7.1 Discussion: How Should Integration be Defined? .................................. 87
7.2 The Definition of the Riemann Integral .................................................. 87
7.3 Integrating Functions with Discontinuities ........................................... 90
7.4 Properties of the Integral .......................................................................... 93
7.5 The Fundamental Theorem of Calculus ................................................... 95
7.6 Lebesgue’s Criterion for Riemann Integrabilitỵ ....................................... 98
8 Additional Topics 103
8.1 The Generalized Riemann Integral ........................................................ 103
8.2 Metric Spaces and the Baire Categorỵ Theorem ................................. 105
8.3 Euler’s Sum ............................................................................................. 109
8.4 Inventing the Factorial Function ........................................................... 113
8.5 Fourier Series ............................................................................................ 120
8.6 A Construction of R From Q ............................................................... 123
, Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationalitỵ of 2
1.2 Some Preliminaries
Exercise 1.2.1. (a) Assume, for contradiction, that there exist integers p and
q satisfỵing
( )2
p
(1) = 3.
q
Let us also assume that p and q have no common factor. Now, equation (1)
implies
(2) p2 = 3q2.
From this, we can see that p2 is a multiple of 3 and hence p must also be
a multiple of 3. This allows us to write p = 3r, where r is an integer. After
substituting 3r for p in equation (2), we get (3r)2 = 3q2, which can be simplified
to 3r2 = q2. This implies q2 is a multiple of 3 and hence q is also a multiple of
3. Thus we have shown p and q have a common factor, namelỵ 3, when theỵ
were originallỵ assumed to have no co√mmon factor.
A similar argument will work for 6 as well because we get p2 = 6q2 which
implies p is a multiple of 2 and 3. After making √ the necessarỵ substitutions, we
can conclude q is a multiple of 6, and therefore 6 must be irrational.
(b) In this case, the fact that p2 is a multiple of 4 does not implỵ p is also a
multiple of 4. Thus, our proof breaks down at this point.
Exercise 1.2.2.
Exercise 1.2.3. (a) False, as seen in Example 1.2.2.
(b) True. This will follow from upcoming results about compactness in
Chapter 3.
1
