Solutions Manual for Understanding Analysis 2nd
Edition by Abbott All Chapters 1-8 Covered
,1 The Real Numbers 1
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Consequences of Completeness .....................................................................................13
1.5 Cardinality........................................................................................................................16
1.6 Cantor’s theorem ........................................................................................................... 23
2 Sequences and Series 27
2.2 The Limit of a Sequence ............................................................................................... 27
2.3 The Algebraic and Order Limit Theorems ..................................................................31
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series ..............39
2.5 Subsequences and the Bolzano–Weierstrass Theorem ................................................46
2.6 The Cauchy Criterion .....................................................................................................51
2.7 Properties of Infinite Series...........................................................................................55
2.8 Double Summations and Products of Infinite Series ..................................................64
3 Basic Topology of R 69
3.2 Open and Closed Sets....................................................................................................69
3.3 Compact Sets..................................................................................................................76
3.4 Perfect Sets and Connected Sets ..................................................................................82
3.5 Baire’s Theorem .............................................................................................................87
4 Functional Limits and Continuity 91
4.2 Functional Limits ............................................................................................................91
4.3 Continuous Functions .....................................................................................................96
4.4 Continuous Functions on Compact Sets ....................................................................105
4.5 The Intermediate Value Theorem...............................................................................112
4.6 Sets of Discontinuity ..................................................................................................... 115
5 The Derivative 119
5.2 Derivatives and the Intermediate Value Property .....................................................119
5.3 The Mean Value Theorems ......................................................................................... 127
5.4 A Continuous Nowhere-Differentiable Function ........................................................ 134
iii
,6 Sequences and Series of Functions 141
6.2 Uniform Convergence of a Sequence of Functions ..................................................... 141
6.3 Uniform Convergence and Differentiation .................................................................. 153
6.4 Series of Functions ........................................................................................................157
6.5 Power Series.................................................................................................................. 162
6.6 Taylor Series ................................................................................................................. 169
6.7 The Weierstrauss Approximation Theorem ............................................................... 176
7 The Riemann Integral 181
7.2 The Definition of the Riemann Integral ..................................................................... 181
7.3 Integrating Functions with Discontinuities.................................................................184
7.4 Properties of the Integral ............................................................................................. 189
7.5 The Fundamental Theorem of Calculus...................................................................... 193
7.6 Lebesgue’s Criterion for Riemann Integrability ......................................................... 199
8 Additional Topics 205
8.1 The Generalized Riemann Integral ............................................................................. 205
8.2 Metric Spaces and the Baire Category Theorem ......................................................209
8.3 Euler’s Sum ................................................................................................................... 217
8.4 Inventing the Factorial Function ................................................................................. 225
8.5 Fourier Series ................................................................................................................ 237
8.6 A Construction of R from Q ...................................................................................... 245
, Chapter 1
The Real Numbers
1.2 Some Preliminaries
Exercise 1.2.1 √ √
(a) Prove that 3 is irrational. Does a similar similar argument work to show 6 is
irrational?
√
(b) Where does the proof break down if we try to prove 4 is irrational?
Solution
(a) Suppose for contradiction that p/q is a fraction in lowest terms, and that (p/q)2 = 3.
Then p2 = 3q2 implying p is a multiple of 3 since 3 is not a perfect square. Therefore
we can write p as 3r for some r, substituting we get (3r)2 = 3q2 and 3r2 = q2 implying
q is √
a l s o a multiple of 3 contradicting the assumption that p/q is in lowest terms.
For 6 the same argument applies, since 6 is not a perfect square.
(b) 4 is a perfect square, meaning p2 = 4q2 does not imply that p is a multiple of four as
p could be 2.
Exercise 1.2.2
Show that there is no rational number satisfying 2r = 3
Solution
If r = 0 clearly 2r = 1 = 3, if r = 0 set p/q = r to get 2p = 3q which is impossible since 2
and 3 share no factors.
Exercise 1.2.3
Decide which of the following represent true statements about the nature of sets. For any
that are false, provide a specific example where the statement in question does not hold.
(a) If A1 ⊇ A2 ⊇ A3T⊇ A4 · · · are all sets containing an infinite number of elements, then
the intersection
(b) If A1 ⊇ A2 T ⊇ A3 ⊇ A4 · · · are all finite, nonempty sets of real numbers, then the
intersection
1
Edition by Abbott All Chapters 1-8 Covered
,1 The Real Numbers 1
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Consequences of Completeness .....................................................................................13
1.5 Cardinality........................................................................................................................16
1.6 Cantor’s theorem ........................................................................................................... 23
2 Sequences and Series 27
2.2 The Limit of a Sequence ............................................................................................... 27
2.3 The Algebraic and Order Limit Theorems ..................................................................31
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series ..............39
2.5 Subsequences and the Bolzano–Weierstrass Theorem ................................................46
2.6 The Cauchy Criterion .....................................................................................................51
2.7 Properties of Infinite Series...........................................................................................55
2.8 Double Summations and Products of Infinite Series ..................................................64
3 Basic Topology of R 69
3.2 Open and Closed Sets....................................................................................................69
3.3 Compact Sets..................................................................................................................76
3.4 Perfect Sets and Connected Sets ..................................................................................82
3.5 Baire’s Theorem .............................................................................................................87
4 Functional Limits and Continuity 91
4.2 Functional Limits ............................................................................................................91
4.3 Continuous Functions .....................................................................................................96
4.4 Continuous Functions on Compact Sets ....................................................................105
4.5 The Intermediate Value Theorem...............................................................................112
4.6 Sets of Discontinuity ..................................................................................................... 115
5 The Derivative 119
5.2 Derivatives and the Intermediate Value Property .....................................................119
5.3 The Mean Value Theorems ......................................................................................... 127
5.4 A Continuous Nowhere-Differentiable Function ........................................................ 134
iii
,6 Sequences and Series of Functions 141
6.2 Uniform Convergence of a Sequence of Functions ..................................................... 141
6.3 Uniform Convergence and Differentiation .................................................................. 153
6.4 Series of Functions ........................................................................................................157
6.5 Power Series.................................................................................................................. 162
6.6 Taylor Series ................................................................................................................. 169
6.7 The Weierstrauss Approximation Theorem ............................................................... 176
7 The Riemann Integral 181
7.2 The Definition of the Riemann Integral ..................................................................... 181
7.3 Integrating Functions with Discontinuities.................................................................184
7.4 Properties of the Integral ............................................................................................. 189
7.5 The Fundamental Theorem of Calculus...................................................................... 193
7.6 Lebesgue’s Criterion for Riemann Integrability ......................................................... 199
8 Additional Topics 205
8.1 The Generalized Riemann Integral ............................................................................. 205
8.2 Metric Spaces and the Baire Category Theorem ......................................................209
8.3 Euler’s Sum ................................................................................................................... 217
8.4 Inventing the Factorial Function ................................................................................. 225
8.5 Fourier Series ................................................................................................................ 237
8.6 A Construction of R from Q ...................................................................................... 245
, Chapter 1
The Real Numbers
1.2 Some Preliminaries
Exercise 1.2.1 √ √
(a) Prove that 3 is irrational. Does a similar similar argument work to show 6 is
irrational?
√
(b) Where does the proof break down if we try to prove 4 is irrational?
Solution
(a) Suppose for contradiction that p/q is a fraction in lowest terms, and that (p/q)2 = 3.
Then p2 = 3q2 implying p is a multiple of 3 since 3 is not a perfect square. Therefore
we can write p as 3r for some r, substituting we get (3r)2 = 3q2 and 3r2 = q2 implying
q is √
a l s o a multiple of 3 contradicting the assumption that p/q is in lowest terms.
For 6 the same argument applies, since 6 is not a perfect square.
(b) 4 is a perfect square, meaning p2 = 4q2 does not imply that p is a multiple of four as
p could be 2.
Exercise 1.2.2
Show that there is no rational number satisfying 2r = 3
Solution
If r = 0 clearly 2r = 1 = 3, if r = 0 set p/q = r to get 2p = 3q which is impossible since 2
and 3 share no factors.
Exercise 1.2.3
Decide which of the following represent true statements about the nature of sets. For any
that are false, provide a specific example where the statement in question does not hold.
(a) If A1 ⊇ A2 ⊇ A3T⊇ A4 · · · are all sets containing an infinite number of elements, then
the intersection
(b) If A1 ⊇ A2 T ⊇ A3 ⊇ A4 · · · are all finite, nonempty sets of real numbers, then the
intersection
1
