**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.*

**Unit 1 Time Advisory**

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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: The article above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/ra/realanal.pdf).
```

**Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 2: Real Numbers and Fields: Sections 11-12: Powers with Arbitrary Real Exponents and Irrationals”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 2: Real Numbers and Fields: Sections 11-12: Powers with Arbitrary Real Exponents and Irrationals” (PDF)Instructions: Please read “Powers with Arbitrary Real Exponents and Irrationals” on pages 46-50.

Terms of Use:

*Mathematical Analysis I*was written by Elias Zakon and relicensed under a Creative Commons-By Attribution 3.0 Unported Licencse as part of the Saylor Foundation’s Open Textbook Challenge. It is attributed to the Trillia Group and Elias Zakon.

**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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.
Terms of Use: The article above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
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**Link: University of California, San Diego: Jiri Lebl’s*Basic Analysis: Introduction to Real Analysis*: “Section 1.3.1: Absolute Value: Exercises”*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.
Terms of Use: The article above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/). It is
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.
Terms of Use: *Mathematical Analysis I* was written by Elias Zakon
and relicensed under a [Creative Commons-By Attribution 3.0 Unported
Licencse](http://creativecommons.org/licenses/by/3.0/) as part of
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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**Activity: University of California, San Diego: Jiri Lebl’s**Link: University of California, San Diego: Jiri Lebl’s*Basic Analysis: Introduction to Real Analysis*: “Exercise 1.4.1: Real Numbers: Intervals and the Size of R: Exercises”*Basic Analysis: Introduction to Real Analysis*: “Exercise 1.4.1: Real Numbers: Intervals and the Size of R: Exercises” (PDF)Work through the problems on pages 36 and 37. 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.