Solutions Manual for:
Understanding Analysis, Second Edition
Stephen Abbott
Middlebury College
June 25, 2015
,Author’s note
What began as a desire to sketch out a simple answer key for the problems
in Understanding Analysis inevitably evolved into something a bit more ambi-
tious. As I was generating solutions for the nearly 200 odd-numbered exercises
in the text, I found myself adding regular commentary on common pitfalls and
strategies that frequently arise. My sense is that this manual should be a use-
ful supplement to instructors teaching a course or to individuals engaged in an
independent study. As with the textbook itself, I tried to write with the in-
troductory student firmly in mind. In my teaching of analysis, I have come to
understand the strong correlation between how students learn analysis and how
they write it. A final goal I have for these notes is to illustrate by example how
the form and grammar of a written argument are intimately connected to the
clarity of a proof and, ultimately, to its validity.
The decision to include only the odd-numbered exercises was a compromise
between those who view access to the solutions as integral to their educational
needs, and those who strongly prefer that no solutions be available because of the
potential for misuse. The total number of exercises was significantly increased
for the second edition, and almost every even-numbered problem (in the regular
sections of the text) is one that did not appear in the first edition. My hope is
that this arrangement will provide ample resources to meet the distinct needs
of these different audiences.
I would like to thank former students Carrick Detweiller, Katherine Ott,
Yared Gurmu, and Yuqiu Jiang for their considerable help with a preliminary
draft. I would also like to thank the readers of Understanding Analysis for the
many comments I have received about the text. Especially appreciated are the
constructive suggestions as well as the pointers to errors, and I welcome more
of the same.
Middlebury, Vermont Stephen Abbott
May 2015
v
,vi Author’s note
, Contents
Author’s note v
1 The Real Numbers √ 1
1.1 Discussion: The Irrationality of 2 . . . . . . . . . . . . . . . . . 1
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 4
1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 6
1.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Properties of Infinite Series . . . . . . . . . . . . . . . . . . . . . 28
2.8 Double Summations and Products of Infinite Series . . . . . . . . 31
3 Basic Topology 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 Continuity 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
Understanding Analysis, Second Edition
Stephen Abbott
Middlebury College
June 25, 2015
,Author’s note
What began as a desire to sketch out a simple answer key for the problems
in Understanding Analysis inevitably evolved into something a bit more ambi-
tious. As I was generating solutions for the nearly 200 odd-numbered exercises
in the text, I found myself adding regular commentary on common pitfalls and
strategies that frequently arise. My sense is that this manual should be a use-
ful supplement to instructors teaching a course or to individuals engaged in an
independent study. As with the textbook itself, I tried to write with the in-
troductory student firmly in mind. In my teaching of analysis, I have come to
understand the strong correlation between how students learn analysis and how
they write it. A final goal I have for these notes is to illustrate by example how
the form and grammar of a written argument are intimately connected to the
clarity of a proof and, ultimately, to its validity.
The decision to include only the odd-numbered exercises was a compromise
between those who view access to the solutions as integral to their educational
needs, and those who strongly prefer that no solutions be available because of the
potential for misuse. The total number of exercises was significantly increased
for the second edition, and almost every even-numbered problem (in the regular
sections of the text) is one that did not appear in the first edition. My hope is
that this arrangement will provide ample resources to meet the distinct needs
of these different audiences.
I would like to thank former students Carrick Detweiller, Katherine Ott,
Yared Gurmu, and Yuqiu Jiang for their considerable help with a preliminary
draft. I would also like to thank the readers of Understanding Analysis for the
many comments I have received about the text. Especially appreciated are the
constructive suggestions as well as the pointers to errors, and I welcome more
of the same.
Middlebury, Vermont Stephen Abbott
May 2015
v
,vi Author’s note
, Contents
Author’s note v
1 The Real Numbers √ 1
1.1 Discussion: The Irrationality of 2 . . . . . . . . . . . . . . . . . 1
1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 4
1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 6
1.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Properties of Infinite Series . . . . . . . . . . . . . . . . . . . . . 28
2.8 Double Summations and Products of Infinite Series . . . . . . . . 31
3 Basic Topology 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 Continuity 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
