Math 127A: Real Analysis, Winter 2019

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Announcements

Solutions to the Final for 127A-A are here.
Solutions to the Final for 127A-B 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 Bolzano-Weierstrass 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 p-series
• 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 (127A-A and 127A-B), each of which has two sections (A01-A02 and B01-B02). 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:
127A-A: Wednesday, March 20, 1:00–3:00 p.m.
127A-B: 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 127A-A are here.
Solutions to Midterm 1 for 127A-B 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 Bolzano-Weierstrass 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 127A-A are here.
Solutions to Midterm 2 for 127A-B 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 p-series
• 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:

• Chapter 1: Sets and Functions

The class will cover Chapters 2–7 of the notes:

• Chapter 2: Numbers
• Chapter 3: Sequences
• Chapter 4: Series
• Chapter 5: Topology of the Real Numbers
• Chapter 6: Limits of Functions
• Chapter 7: Continuous Functions

Supplementary Notes

Here are some additional notes for topics not covered in the Lecture Notes.

• Density of the rationals
• The liminf and limsup of bounded sequences

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, 1.2.10

  Set 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, 2.4.5

  Set 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, 2.6.7

  Set 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, 2.7.13

  Set 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, 3.3.8

  Set 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), 4.3.9

  Set 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