ALL 8 CHAPTERS COVERED
SOLUTIONS MANUAL
, INSTRUCTORS’
SOLUTIONS
MANUAL
for
A First Course in
Mathematical Analysis
David A. Brannan
© David A Brannan 2015
1
,INTRODUCTION
This manual contains solutions to the end-of-chapter Exercises in the book A First Course in
Mathematical Analysis by David Alexander Brannan, first published in 2006 by Cambridge
University Press as ISBN: 0-521-68424-2, and reprinted with corrections in 2012.
Note that each solution in this manual is not necessarily the only possible solution to that exercise
– generally there are several possible solutions, each equally valid. For many exercises, we offer
a relevant place in the text to which instructors can helpfully refer students who find the exercise
difficult – so that they can revise that topic.
Students often find mathematics hard, especially Mathematical Analysis. This textbook has been
written in a particular style to try to address many of the common problems; it can be used either
as a set text for a lecture course, or as a backup reference for a lecture course, or for self-study.
It is important that students tackle a reasonable number of the end-of-chapter exercises. Some
exercises are straight-forward, while others are more stetching. Students should remember to
read all of the text (except perhaps those portions marked as ‘optional’) carefully before attempting
the exercises; and, if they get into difficulty with an exercise, they should refer back to the text and
its examples and problems to see if there is something similar there that they can use as a model
or as inspiration.
Students should be encouraged to use diagrams when studying mathematics! In the case of
Mathematical Analysis, however, they need to remember that diagrams can be both helpful and
misleading:
A well-chosen diagram can shed light on a situation, suggesting a way forward to solve an
exercise, but
Some diagrams can be misleading, suggesting incorrect results or overlooking situations that
can occur.
Even professional mathematicians can be mislead by a diagram! So, while students should be
strongly encouraged to draw diagrams to illustrate situations, they should be strongly warned not
to jump too quickly to conclusions from their diagrams!
Many students have found the following book useful in helping them to read mathematics, study
mathematics, and do mathematics:
Kevin Houston, How to Think Like a Mathematician: A Companion to Undergraduate
Mathematics, Cambridge University Press, Cambridge (2009); ISBN: 9780521719780 (pbk).
Every effort has been made to avoid errors and typos. If any errors/typos remain (and surely there
are some!), the author would be grateful to be told about them at the email address:
The author would also welcome any comments or suggestions for improvements, and would aim
to reply to all messages.
2
, CONTENTS
Solutions to the end-of-chapter Exercises:
1 Numbers
1.1 Real numbers 4
1.2 Inequalities 5
1.3 Proving inequalities 5
1.4 Least upper bounds and greatest lower bounds 7
2 Sequences
2.1 Introducing sequences 10
2.2 Null sequences 10
2.3 Convergent sequences 12
2.4 Divergent sequences 14
2.5 The Monotone Convergence Theorem 16
3 Series
3.1 Introducing series 21
3.2 Series with non-negative terms 22
3.3 Series with positive and negative terms 25
4 Continuity
4.1 Continuous functions 28
4.2 Properties of continuous functions 32
4.3 Inverse functions 34
5 Limits and continuity
5.1 Limits of functions 36
5.2 Asymptotic behaviour of functions 38
5.3 Limits of functions ─ using ε and δ 40
5.4 Continuity ─ using ε and δ 41
5.5 Uniform continuity 43
6 Differentiation
6.1 Differentiable functions 45
6.2 Rules for differentiation 46
6.3 Rolle’s Theorem 48
6.4 The Mean Value Theorem 50
6.5 L’Hôpital’s Rule 53
7 Integration
7.1 The Riemann integral 56
7.2 Properties of integrals 60
7.3 Fundamental Theorem of Calculus 61
7.4 Inequalities for integrals and their applications 64
7.5 Stirling’s Formula for 𝑛! 67
8 Power series
8.1 Taylor polynomials 69
8.2 Taylor’s Theorem 70
8.3 Convergence of power series 73
8.4 Manipulating power series 75
3
SOLUTIONS MANUAL
, INSTRUCTORS’
SOLUTIONS
MANUAL
for
A First Course in
Mathematical Analysis
David A. Brannan
© David A Brannan 2015
1
,INTRODUCTION
This manual contains solutions to the end-of-chapter Exercises in the book A First Course in
Mathematical Analysis by David Alexander Brannan, first published in 2006 by Cambridge
University Press as ISBN: 0-521-68424-2, and reprinted with corrections in 2012.
Note that each solution in this manual is not necessarily the only possible solution to that exercise
– generally there are several possible solutions, each equally valid. For many exercises, we offer
a relevant place in the text to which instructors can helpfully refer students who find the exercise
difficult – so that they can revise that topic.
Students often find mathematics hard, especially Mathematical Analysis. This textbook has been
written in a particular style to try to address many of the common problems; it can be used either
as a set text for a lecture course, or as a backup reference for a lecture course, or for self-study.
It is important that students tackle a reasonable number of the end-of-chapter exercises. Some
exercises are straight-forward, while others are more stetching. Students should remember to
read all of the text (except perhaps those portions marked as ‘optional’) carefully before attempting
the exercises; and, if they get into difficulty with an exercise, they should refer back to the text and
its examples and problems to see if there is something similar there that they can use as a model
or as inspiration.
Students should be encouraged to use diagrams when studying mathematics! In the case of
Mathematical Analysis, however, they need to remember that diagrams can be both helpful and
misleading:
A well-chosen diagram can shed light on a situation, suggesting a way forward to solve an
exercise, but
Some diagrams can be misleading, suggesting incorrect results or overlooking situations that
can occur.
Even professional mathematicians can be mislead by a diagram! So, while students should be
strongly encouraged to draw diagrams to illustrate situations, they should be strongly warned not
to jump too quickly to conclusions from their diagrams!
Many students have found the following book useful in helping them to read mathematics, study
mathematics, and do mathematics:
Kevin Houston, How to Think Like a Mathematician: A Companion to Undergraduate
Mathematics, Cambridge University Press, Cambridge (2009); ISBN: 9780521719780 (pbk).
Every effort has been made to avoid errors and typos. If any errors/typos remain (and surely there
are some!), the author would be grateful to be told about them at the email address:
The author would also welcome any comments or suggestions for improvements, and would aim
to reply to all messages.
2
, CONTENTS
Solutions to the end-of-chapter Exercises:
1 Numbers
1.1 Real numbers 4
1.2 Inequalities 5
1.3 Proving inequalities 5
1.4 Least upper bounds and greatest lower bounds 7
2 Sequences
2.1 Introducing sequences 10
2.2 Null sequences 10
2.3 Convergent sequences 12
2.4 Divergent sequences 14
2.5 The Monotone Convergence Theorem 16
3 Series
3.1 Introducing series 21
3.2 Series with non-negative terms 22
3.3 Series with positive and negative terms 25
4 Continuity
4.1 Continuous functions 28
4.2 Properties of continuous functions 32
4.3 Inverse functions 34
5 Limits and continuity
5.1 Limits of functions 36
5.2 Asymptotic behaviour of functions 38
5.3 Limits of functions ─ using ε and δ 40
5.4 Continuity ─ using ε and δ 41
5.5 Uniform continuity 43
6 Differentiation
6.1 Differentiable functions 45
6.2 Rules for differentiation 46
6.3 Rolle’s Theorem 48
6.4 The Mean Value Theorem 50
6.5 L’Hôpital’s Rule 53
7 Integration
7.1 The Riemann integral 56
7.2 Properties of integrals 60
7.3 Fundamental Theorem of Calculus 61
7.4 Inequalities for integrals and their applications 64
7.5 Stirling’s Formula for 𝑛! 67
8 Power series
8.1 Taylor polynomials 69
8.2 Taylor’s Theorem 70
8.3 Convergence of power series 73
8.4 Manipulating power series 75
3
