Course information
Instructor: Prof. John Hunter
Lectures:
127AA. MWF 11:00–11:50 a.m., Wellman 202
127AB. MWF 2:10–3:00 p.m., Olson 158
Office: MSB 3230
Office hours: MW 3:30–5:00 p.m.
email: jkhunter@ucdavis.edu
Discussions and TA for 127AA
Discussion sections:
A01. Thur 5:10–6:00 p.m., Hutchinson 102
A02. Thur 6:10–7:00 p.m., Hutchinson 102
TA: Joshua Sumpter
Office: MSB 2127
Office hours: TR 1:00–2:00 p.m.
Discussions and TA for 127AB
Discussion sections:
B01. Tue 5:10–6:00 p.m., Olson 101
B02. Tue 6:10–7:00 p.m., Olson 101
TA: Yunshen Zhou
Office: MSB 3131
Office hours: R 3:00–5:00 p.m.
Academic Conduct:
The UC Davis Code of Academic Conduct is here.Announcements
Solutions to the Final for 127AA are here.
Solutions to the Final for 127AB are here.
The final will cover all of the material in the course.
 Algebraic and order properties of the real numbers
 Definition of the supremum and infimum of sets and their properties
 Dedekind completeness axiom for the real numbers
 Finite intersection and Archimedean properties of the real numbers
 Density of the rationals in the reals
 Sequences, convergence, and limits
 Divergence of sequences to infinity
 Algebraic and order properties of limits
 Convergence of bounded monotone sequences
 Definition of the limsup and liminf of a bounded sequence
 Relations between the limsup, liminf, and the limit of a sequence
 Cauchy sequences and their convergence
 Subsequences and the BolzanoWeierstrass theorem
These topics are covered in Sections 1.1–1.4 and 2.1–2.6 of the text, and Chapters 2–3 of the course lecture notes.
 Convergence of series
 Geometric, telescoping, and pseries
 Series of positive terms
 Cauchy condition for series
 Absolutely and conditionally convergent series
 The comparison, ratio, and root tests
 The alternating series test
 Rearrangements of series
 Definitions of open and closed sets
 Properties of open and closed sets
 Limit, isolated, and interior points of a set
 Closure and interior of a set
 Compact sets
These topics are covered in Sections 2.4, 2.7, and 3.2–3.3 of the text and in Sections 4.1–4.4, 4.6–4.8, and 5.1–5.4 of the course lecture notes.
 Limits of functions
 Properties of functional limits
 Continuous functions
 Uniform continuity
 Continuous functions on compact sets
 Continuous functions and open sets
 Intermediate value theorem
These topics are covered in Sections 4.2–4.5 of the text and in Sections 6.1–6.3 and 7.1–7.6 of the course lecture notes.
A summary of the main definitions and theorems covered in the course is here. There are no proofs or examples; you should think about both as you review the summary. Let me know if you find any errors or significant omissions.
Course Homegage
This is the course homepage for MAT 127A, Winter 2019. Exam scores and announcements will be posted on Canvas, but all other course information will be posted here.
Please note that I'm teaching two 127A classes in winter quarter (127AA and 127AB), each of which has two sections (A01A02 and B01B02). This webpage is the homepage for both classes. Any differences in course information between the classes are noted explicitly below.
Important Dates
 Instruction begins: Monday, January 7
 Last day to add: Wednesday, January 23
 Last day to drop: Monday, February 4
 Last class: Friday, March 15
 Academic holidays: Monday, January 21; Monday, February 18
Exams
There will be two Midterms and a Final,
Midterm 1:
Friday, February 1 (in class)
Midterm 2:
Wednesday, February 27 (in class)
Final:
127AA: Wednesday, March 20, 1:00–3:00 p.m.
127AB: Thursday, March 21, 8:00–10:00 a.m.
Homework
Homework will be assigned weekly from the text, and a hard copy will be due in class on Fridays. Please write neatly, or type, and staple your solutions. In addition, put your name and section (A01, A02,B01, B02) on your homework. Homework will be returned in the discussion sections.
Proofs and answers should be written in clear, grammatically correct, complete sentences (as in the text or class notes) with full justification of your reasoning.
The standard tool for writing mathematical papers is LaTeX, or one of its many variants. If you want to learn how to use it, see here to get started. (This suggestion is only if you're interested  you're not required to use latex for your homework.)
Midterm 1
Solutions to Midterm 1 for 127AA are here.
Solutions to Midterm 1 for 127AB are here.
Midterm 1 will be in class on Friday, February 1. An outline of the topics is as follows:
 Algebraic and order properties of the real numbers
 Definition of the supremum and infimum of sets and their properties
 Dedekind completeness axiom for the real numbers
 Finite intersection and Archimedean properties of the real numbers
 Density of the rationals in the reals
 Sequences, convergence, and limits
 Divergence of sequences to infinity
 Algebraic and order properties of limits
 Convergence of bounded monotone sequences
 Definition of the limsup and liminf of a bounded sequence
 Relations between the limsup, liminf, and the limit of a sequence
 Cauchy sequences and their convergence
 Subsequences and the BolzanoWeierstrass theorem
These topics are covered in Sections 1.1–1.4 and 2.1–2.6 of the text, and Chapters 2–3 of the course lecture notes.
Some sample exam questions are here.
Midterm 2
Solutions to Midterm 2 for 127AA are here.
Solutions to Midterm 2 for 127AB are here.
Midterm 2 is in class on Wednesday, February 27. It will cover series (starting with the material since Midterm 1) and the topology of R. An outline of the topics is as follows:
 Convergence of series
 Geometric, telescoping, and pseries
 Series of positive terms
 Cauchy condition for series
 Absolutely and conditionally convergent series
 The comparison, ratio, and root tests
 The alternating series test
 Rearrangements of series
 Definitions of open and closed sets
 Properties of open and closed sets
 Limit, isolated, and interior points of a set
 Closure and interior of a set
These topics are covered in Sections 2.4, 2.7, and 3.2 of the text and in Sections 4.1–4.4, 4.6–4.8, 5.1–5.2 of the course lecture notes.
Some sample problems are here.
I'll hold a review session for the midterm on Tue, Feb 26, 4.10–5 p.m. in 176 Everson.
Grade
The course grade will based on exams and homework, weighted as follows:
 15%: Homework
 25%: Each Midterm
 35%: Final
Required Text
Understanding Analysis, Stephen Abbott, Second Edition, 2015.
This class will cover the first four chapters of the text:
 Chapter 1: The Real Numbers
 Chapter 2: Sequences and Series
 Chapter 3: Topology of the Real Numbers
 Chapter 4: Functional Limits and Continuity
The department syllabus and list of topics for the class is here.
Lecture Notes
In addition to the text, I will use my own lecture notes available online here:
An Introduction to Real Analysis
Background material is in:
The class will cover Chapters 2–7 of the notes:
Supplementary Notes
Here are some additional notes for topics not covered in the Lecture Notes.
Problem Sets
Problem numbers refer to the exercises in the text (2nd edition).
Homework is due in class on Fridays. Please staple your solutions
and write your section number.

Set 1 (Fri, Jan 11)
Sec 1.2, p. 11: 1.2.1, 1.2.3, 1.2.4, 1.2.6, 1.2.8, 1.2.9, 1.2.10Set 2 (Fri, Jan 18)
Sec 1.3, p. 18: 1.3.2, 1.3.3, 1.3.5, 1.3.9
Sec 1.4, p. 24: 1.4.4, 1.4.5, 1.4.7
Sec 2.2, p. 47: 2.2.2, 2.2.4
Sec 2.3, p. 54: 2.3.1, 2.3.3, 2.3.6, 2.3.10(a)(b)(d)
A solution to 1.3.3 is here.Set 3 (Fri, Jan 25)
Sec 2.3, p. 54: 2.3.11, 2.3.12
Sec 2.4, p. 59: 2.4.1, 2.4.2, 2.4.3, 2.4.4, 2.4.5Set 4 (Fri, Feb 1)
Sec 2.4, p. 61: 2.4.7
Sec 2.5, p. 65: 2.5.1, 2.5.2, 2.5.7, 2.5.9
Sec 2.6, p. 70: 2.6.2, 2.6.4, 2.6.7Set 5 (Fri, Feb 8)
Sec 2.7, p. 76: 2.7.2, 2.7.3, 2.7.4, 2.7.5, 2.7.7, 2.7.8, 2.7.9, 2.7.12, 2.7.13Set 6 (Fri, Feb 15)
Sec 2.4, p. 59: 2.4.10
Sec 2.7, p. 76: 2.7.1, 2.7.10, 2.7.14
Read Sections 2.8–2.9 in the text (this material won't be covered in lectures)Set 7 (Fri, Feb 22)
Sec 3.2, p. 93: 3.2.1–3.2.15 (all of the problems)Set 8 (Fri, Mar 1)
Sec 3.3, p. 99: 3.3.1, 3.3.2, 3.3.4, 3.3.5, 3.3.8Set 9 (Fri, Mar 8)
Sec 3.3, p. 99: 3.3.9, 3.3.11
Sec 4.2, p. 120: 4.2.5, 4.2.6, 4.2.7, 4.2.9, 4.2.10, 4.2.11
Sec 4.3, p. 126: 4.3.1, 4.3.3, 4.3.5, 4.3.6, 4.3.7(a), 4.3.9Set 10 (Fri, Mar 15)
Sec 4.4, p. 134: 4.4.1, 4.4.2, 4.4.4, 4.4.6, 4.4.7, 4.4.8, 4.4.9, 4.4.11
Sec 4.5, p. 139: 4.5.2, 4.5.3, 4.5.5, 4.5.7