# MA241: Real Analysis I

Unit 1: The Real Number System   In this unit, we will learn or be reminded of the tools of set theory, which underlie rigorous mathematical proof, before investigating the real numbers as a field. Beginning with the rational numbers, we will construct the real number system, a project which took mathematicians hundreds of years to fully justify. Along the way, we will encounter several fundamental concepts, including the well-ordering principle, the completeness axiom, and the Archimedean property.

We will see the proof of the existence of irrational numbers and learn how to use proof by induction. Finally, we will see proven a number of results about the cardinality of sets.

This unit should take you approximately 33 hours to complete.

☐    Subunit 1.1: 3 hours

☐    Subunit 1.2: 2 hours

☐    Subunit 1.3: 4 hours

☐    Subunit 1.4: 5 hours

☐    Subunit 1.5: 2 hours

☐    Subunit 1.6: 4 hours

☐    Subunit 1.7: 2 hours

☐    Subunit 1.8: 2 hours

☐    Subunit 1.9: 4 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- use set notation and quantifiers correctly in mathematical statements and proofs; - use proof by induction when appropriate; - use proof by contradiction when appropriate; - define the rational numbers, the natural numbers, and the real numbers, and understand their relationship to one another; - define the well-ordering principle, the completeness/supremum property of the real line, and the Archimedean property; - prove the existence of irrational numbers; - state the axioms of the real numbers, and use them to justify true statements; - define supremum and infimum; - correctly and fluently perform algebraic operations on expressions involving absolute value, and state the triangle inequality; - define and identify injective, surjective, and bijective mappings; - and name the various cardinalities of sets and identify the cardinality of a given set.

1.1 Sets, Operations on Sets, and Quantifiers   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 1-3: Sets and Operations on Sets; Quantifiers” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 1-3: Sets and Operations on Sets; Quantifiers” (PDF)

`````` Instructions: Please read the indicated sections, on pages 1-6.
Please note that you will be returning to this resource throughout
the course, so you may prefer to save the PDF to your desktop for
quick reference.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 1: Constructing the Rational Numbers” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 1: Constructing the Rational Numbers” (YouTube)

Instructions: Please watch this lecture, in which Professor Su discusses set and function/relation notation. He also discusses the historical development of the study of analysis and construct (that is, rigorously justify from first principles) the rational numbers.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

1.2 Axioms of the Real Numbers   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2 “Real Numbers and Fields: Sections 1-4: Axioms and Basic Definitions” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 1-4: Axioms and Basic Definitions” (PDF)

`````` Instructions: Please read Sections 1-4, “Axioms and Basic
Definitions,” on pages 23-27.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

1.3 Integers and the Rational Numbers   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 7: Integers and Rationals” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 7: Integers and Rationals” (PDF)

`````` Instructions: Please read Section 7, “Integers and Rationals,” on
pages 34-36. Note: The reading for this section focuses on integers
and rationals, but one cannot ponder the rational numbers without
considering the existence of irrationals. Hence, in the video
lectures for this section, the existence of irrationals will be
proven. The readings on the irrational numbers come in subunit 1.4
after a few more concepts have been developed. This difference in
ordering should not interfere with your understanding of the
lectures for this section.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 2: Properties of Q” and “Lecture 3: Construction of the Real Numbers” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 2: Properties of Q” (YouTube) and “Lecture 3: Construction of the Real Numbers” (YouTube)

Instructions: Please watch both of these videos. In the first lecture, Professor Su discusses the rational numbers in further detail, defining addition and multiplication, and he will use them to introduce important concepts such as ordering. He proves the existence of irrationals, such as the square root of 2. In the second lecture, Professor Su discusses Dedekind cuts, the least upper bound property of the reals (a.k.a. the completeness property). Feel free to end the lecture at 42:00 (the discussion of Dedekind cuts).

Watching these videos and pausing to take notes should take approximately 2 hours and 30 minutes.

1.4 Upper and Lower Bounds   1.4.1 Upper and Lower Bounds   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 8-9: Upper and Lower Bounds and Completeness” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 8-9: Upper and Lower Bounds and Completeness” (PDF)

`````` Instructions: Please read Sections 8-9, “Upper and Lower Bounds and
Completeness,” on pages 36-40.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

1.4.2 The Archimedean Property   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 10: Some Consequences of the Completeness Axiom” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Section 10: Some Consequences of the Completeness Axiom” (PDF)

`````` Instructions: Please read “Some Consequences of the Completeness
Axiom” on pages 43-46.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 4: The Least Upper Bound Property” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 4: The Least Upper Bound Property” (YouTube)

Instructions: Please watch this video. Please note that this material covers the topics outlined in sub-subunits 1.4.1 and 1.4.2. In this lecture, Professor Su further discusses Dedekind cuts and the existence of arbitrary real powers of rationals. He goes through the least upper bound property of the reals (a.k.a. the completeness property), the greatest lower bound property, and the Archimedean Property, which he proves. He proves the density of the rationals in the real line. He also gives properties of the supremum. Mastery of these topics is essential to developing a thorough understanding of analysis.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

1.5 Irrationals   - Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Exercise 1.2.5: Real Numbers: The Set of Real Numbers: Exercises” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Exercise 1.2.5: Real Numbers: The Set of Real Numbers: Exercises” (PDF)

`````` Instructions: Please work through the exercises on page 30 of this
PDF. Because this resource is used by some institutions of higher
learning for the purposes of assigning grades and credit for
classes, complete solutions to the problems are unavailable.
However, it is to your benefit to attempt the problems; your
solutions should mimic the style of the proofs given in the
preceding chapter.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````

1.6 The Natural Numbers and Induction   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 5-6: Natural Numbers and Induction” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 5-6: Natural Numbers and Induction” (PDF)

`````` Instructions: Please read the “Natural Numbers and Induction”
section on pages 27-32.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 6: Principle of Induction” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 6: Principle of Induction” (YouTube)

Instructions: Please watch this video. In this lecture, Professor Su justifies and demonstrates the principle of proof by induction. He shows that it is equivalent to the well-ordering principle.

Watching this lecture and pausing to take notes should take approximately 1 hour and 30 minutes.

1.7 Absolute Value   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3: Real Numbers: Absolute Value” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Section 1.3: Real Numbers: Absolute Value” (PDF)

`````` Instructions: Please read Section 1.3, “Absolute Value,” on pages
31-34. The most important fact we will encounter in this section,
which we will use over and over again in this course and all those
which follow, is the triangle inequality.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````
• Activity: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.3.1: Absolute Value: Exercises” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Section 1.3.1: Absolute Value: Exercises” (PDF)

Instructions: Work through the exercises on page 34. Because this resource is used by some institutions of higher learning for the purposes of assigning grades and credit for classes, complete solutions to the problems are unavailable. However, it is to your benefit to attempt the problems; your solutions should mimic the style of the proofs given in the preceding chapter.

Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0. It is attributed to Jiri Lebl and the original version can be found here.

1.8 Intervals and the Size of R   - Reading: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis: “Section 1.4: Intervals and the Size of R” Link: University of California, San Diego: Jiri Lebl’s Basic Analysis: Introduction to Real Analysis“Section 1.4: Intervals and the Size of R” (PDF)

`````` Instructions: Please read the indicated section, on pages 35 and
36.

attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
``````

1.9 Sets and Countable Sets   1.9.1 Sets: Relations; Mappings   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Sections 4-7: Relations; Mappings” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I“Chapter 1: Set Theory: Sections 4-7: Relations; Mappings” (PDF)

`````` Instructions: Please read “Relations; Mappings” on pages 8-14.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````

1.9.2 Countable Sets   - Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 1: Set Theory: Section 9: Some Theorems on Countable Sets” Link: The Trillia Group: Elias Zakon’s Mathematical Analysis I“Chapter 1: Set Theory: Section 9: Some Theorems on Countable Sets” (PDF)

`````` Instructions: Please read “Some Theorems on Countable Sets” on
pages 18-21.

the Saylor Foundation’s Open Textbook Challenge. It is attributed to
the Trillia Group and Elias Zakon.
``````
• Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 7: Countable and Uncountable Sets” and “Lecture 8: Cantor Diagonalization and Metric Spaces” Link: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 7: Countable and Uncountable Sets” (YouTube) and “Lecture 8: Cantor Diagonalization and Metric Spaces” (YouTube)

Instructions: Please note that these lectures cover the topics outlined in sub-subunits 1.9.1 and 1.9.2 of this course. Please watch each video lecture. Only watch Lecture 8 up to the 51-minute mark.

In “Lecture 7: Countable and Uncountable Sets,” Professor Su revisits functions and relations between sets and important concepts such as “one-to-one/injective,” “onto/surjective,” and “bijective.” He defines finite, infinite, countable, and uncountable sets and power sets, and he discusses the cardinality of certain important sets. In “Lecture 8: Cantor Diagonalization and Metric Spaces,” Professor Su discusses cardinality and Cantor’s diagonalization argument in more detail.

Watching these videos and pausing to take notes should take approximately 2 hours and 30 minutes.