**Unit 2: Metric Spaces**
*In this unit, we will learn about various topological notions and will
be introduced to the more abstract notion of a metric space. A metric is
a function which takes two points in a certain set and tells how “far
apart” they are and which satisfies three special requirements. The
metric can be used to define many different properties of the set to
which it applies (the* metric space*). All of the concepts which you
learned in calculus, especially limits, can be understood and extended
in the context of metric spaces.*

**Unit 2 Time Advisory**

This unit should take you approximately 33 hours to complete.

☐ Subunit 2.1: 18 hours

☐ Subunit 2.2: 4 hours

☐ Subunit 2.3: 7.5 hours

☐ Subunit 2.4: 3.5 hours

**Unit2 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- define metric spaces;
- define open, closed, and bounded sets;
- define cluster points;
- define convergence of sequences and prove or disprove the
convergence of given sequences;
- prove and use properties of limits;
- define density;
- prove standard results about closures, intersections, and unions of
open and closed sets;
- define compactness using both open covers and sequences;
- state and prove the Heine-Borel Theorem;
- state the Bolzano-Weierstrass Theorem;
- state and use the Cantor Finite Intersection Property;
- define Cauchy sequence and prove that specific sequences are Cauchy;
- define completeness;
- show that convergent sequences are Cauchy;
- define limit superior and limit inferior;
- prove that the set of real numbers, equipped with the standard
metric, is complete;
- define convergence of series using the Cauchy criterion;
- and use the comparison, ratio, and root tests to show convergence of
series.

**2.1 Metric Spaces**
**2.1.1 Definition**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 11: Metric
Spaces”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 11: Metric
Spaces”
(PDF)

```
Instructions: Please read the “Metric Spaces” section on pages
95-98.
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 8: Cantor Diagonalization and Metric Spaces”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 8: Cantor Diagonalization and Metric Spaces” (YouTube)Instructions: Please watch the video from the 51-minute mark to the end. In this lecture, Professor Su defines metric spaces and gives examples. He defines open balls and gives examples of open balls in a variety of metrics.

Watching this lecture and pausing to take notes should take approximately 45 minutes.

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

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Problems on Metric Spaces”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces and Metric Spaces: Section 11: Problems on Metric Spaces” (PDF)Instructions: Please scroll down to page 98, and work through problems 1, 9, 11, and 12.

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.

**2.1.2 Open and Closed Sets**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 12: Open and
Closed Sets; Neighborhoods”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 12: Open and
Closed Sets;
Neighborhoods”
(PDF)

```
Instructions: Please read the “Open and Closed Sets; Neighborhoods”
section on pages 101-106.
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.
```

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Problems on Neighborhoods, Open and Closed Sets”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces and Metric Spaces: Section 12: Problems on Neighborhoods, Open and Closed Sets” (PDF)Instructions: Please scroll down to page 107, and work through problems 7, 8, 16, 17, and 18.

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.

**2.1.3 Bounded Sets**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 13: Bounded
Sets; Diameters”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 13: Bounded
Sets;
Diameters”
(PDF)

```
Instructions: Please read the “Bounded Sets; Diameters” section on
pages 108-112.
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.
```

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Problems on Boundedness and Diameters”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces and Metric Spaces: Section 13: Problems on Boundedness and Diameters” (PDF)Instructions: Please scroll down to page 112, and work through problems 2, 4, 11, and 12.

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.

**2.1.4 Cluster Points**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 14: Cluster
Points; Convergent Sequences”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 14: Cluster
Points; Convergent
Sequences”
(PDF)

```
Instructions: Please read the “Cluster Points; Convergent
Sequences” section on pages 114-118.
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 9: Limit Points” and “Lecture 15: Convergence of Sequences”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 9: Limit Points” (YouTube) and “Lecture 15: Convergence of Sequences” (YouTube)Instructions: Please watch these lectures. In the first lecture, Professor Su defines limit (cluster) points and goes through many examples. He also defines interior points, open sets, closed sets, and closures. In the second lecture, Professor Su defines what it means for a sequence to converge.

Watching these lectures 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: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces; Metric Spaces: Section 14: Problems on Cluster Points and Convergence”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces; Metric Spaces: Section 14: Problems on Cluster Points and Convergence” (PDF)Instructions: Please scroll down to page 118, and work through problems 5 and 10.

*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.

**2.1.5 Operations on Convergent Sequences**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 15: Operations
on Convergent Sequences”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 15: Operations
on Convergent
Sequences”
(PDF)

```
Instructions: Please read the indicated section on pages 120-123.
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 16: Subsequences, Cauchy Sequences”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 16: Subsequences, Cauchy Sequences” (YouTube)Instructions: Please watch this lecture through the 30-minute mark. In this lecture, Professor Su further explores properties of limits of sequences.

Watching this lecture and pausing to take notes should take approximately 45 minutes.

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

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Problems on Limits of Sequences”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces and Metric Spaces: Section 15: Problems on Limits of Sequences” (PDF)Instructions: Please scroll down to page 123, and work through problems 1, 10, 12, 13, and 25.

*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.

**2.1.6 Closed Sets and Density**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 16: More on
Cluster Points and Closed Sets; Density”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 16: More on
Cluster Points and Closed Sets;
Density”
(PDF)

```
Instructions: Please read the “More on Cluster Points and Closed
Sets; Density” section on pages 135-139.
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 10: The Relationship Between Open and Closed Sets”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 10: The Relationship Between Open and Closed Sets” (YouTube)Instructions: Please watch this lecture, in which Professor Su revisits the definition of open and closed sets and some of the subtleties involved in manipulating them. He proves standard results, such as that the closure of a set is closed and that a set is closed if and only if its complement is open. He also investigates unions and intersections of open and closed sets. He defines what it means for one set to be dense in another set.

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.

**Activity: The Trillia Group: Elias Zakon’s Mathematical Analysis I: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: Problems on Cluster Points, Closed Sets, and Density”**Link: The Trillia Group: Elias Zakon’s*Mathematical Analysis I*: “Chapter 3: Vector Spaces and Metric Spaces: Section 16: Problems on Cluster Points, Closed Sets, and Density” (PDF)Instructions: Please scroll down to page 140, and work through problems 9, 12, and 17.

*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.

**2.2 Compactness**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 4: Function Limits and Continuity: Section 6: Compact Sets
and Section 7: More on Compactness”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 4: Function Limits and Continuity: Section 6: Compact Sets
and Section 7: More on
Compactness”
(PDF)

```
Instructions: Please read the indicated sections on pages 186-189
and pages 192-194.
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 11: Compact Sets” and “Lecture 12: The Relationship of Compact Sets to Closed Sets”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 11: Compact Sets” (YouTube) and “Lecture 12: The Relationship of Compact Sets to Closed Sets” (YouTube)Instructions: Click on the links above, and watch these lectures. In the first lecture, Professor Su defines compactness from the topological perspective (

*e.g.,*the way it is defined in the reading of Zakon’s Section 4.7 – every open cover of the set must have a finite subcover). This is because in his course he develops this concept before discussing the convergence of sequences. He also (in essence) defines what it means for a set to be relatively open with respect to another set. He proves that compact sets are bounded in Euclidean space. In the second lecture, Professor Su proves that compact sets are closed in Euclidean space. He proves that nested closed intervals in R have nonempty intersection. He also proves that R is uncountable.Also, note that Professor Su will touch on sequential compactness in the lecture for sub-subunit 2.3.2. In this lecture and the one that follows, he uses the open-cover definition.

Watching these lecture sand pausing to take notes should take 2 hours and 45 minutes.Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 13: Compactness and the Heine-Borel Theorem”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 13: Compactness and the Heine-Borel Theorem” (YouTube)Instructions: Please watch this lecture, in which Professor Su proves that closed, bounded intervals on the real line are compact. He then proves the Heine-Borel Theorem (this is exercise 10 in section 4.6 of Zakon’s book). He states a version of the Bolzano-Weierstrass Theorem (which will be discussed in the reading under sub-subunit 2.3.1 for this course) and the Cantor Finite Intersection Property.

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

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

**2.3 Subsequences, Cauchy Sequences, and Completeness**
**2.3.1 Cauchy Sequences and Completeness**
- **Reading: The Trillia Group: Elias Zakon’s Mathematical Analysis I:
“Chapter 3: Vector Spaces and Metric Spaces: Section 17: Cauchy
Sequences; Completeness”**
Link: The Trillia Group: Elias Zakon’s *Mathematical Analysis I*:
“Chapter 3: Vector Spaces and Metric Spaces: Section 17: Cauchy
Sequences;
Completeness”
(PDF)

```
Instructions: Please read the “Cauchy Sequences; Completeness”
section on pages 141-144.
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.
```

**2.3.2 Limits Superior and Inferior and the Bolzano-Weierstrass
Theorem**
- **Reading: University of California, San Diego: Jiri Lebl’s Basic
Analysis: Introduction to Real Analysis: “Section 2.3: Limit
Superior, Limit Inferior, and Bolzano-Weierstrass”**
Link: University of California, San Diego: Jiri Lebl’s

*Basic Analysis: Introduction to Real Analysis*: “Section 2.3: Limit Superior, Limit Inferior, and Bolzano-Weierstrass” (PDF)

```
Instructions: Please read “Limit Superior, Limit Inferior, and
Bolzano-Weierstrass” on pages 61-66.
Terms of Use: The article above is released under [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0
License](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).
```

**Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 16: Subsequences, Cauchy Sequences” and “Lecture 17: Complete Spaces”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 16: Subsequences, Cauchy Sequences” (YouTube) and “Lecture 17: Complete Spaces” (YouTube)Instructions: Please watch both lectures. Watch “Lecture 16: Subsequences, Cauchy Sequences” from the 30-minute mark to the end; watch all of “Lecture 17: Complete Spaces”. In the first lecture, Professor Su defines subsequences and proves several important results about them. Note the definition of sequential compactness at time 43:50. He proves the Bolzano-Weierstrass Theorem. He defines Cauchy sequence and completeness. In the second lecture, Professor Su proves that compact metric spaces are complete. He also proves that Euclidean space is complete. He constructs the completion of a metric space. He discusses bounded sequences and monotonic sequences and proves that bounded, monotonic sequences converge. He defines limit superior and limit inferior and proves several results about them.

Watching these lectures and pausing to take notes should take approximately 1 hour and 45 minutes.

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

**2.4 Series**
- **Reading: University of California, San Diego: Jiri Lebl’s Basic
Analysis: Introduction to Real Analysis: “Section 2.5: Series”**
Link: University of California, San Diego: Jiri Lebl’s

*Basic Analysis: Introduction to Real Analysis*: “Section 2.5: Series” (PDF)

```
Instructions: Please read “Series” on pages 72-81.
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).
```

**Lecture: YouTube: Harvey Mudd College: Professor Francis Su’s Real Analysis: “Lecture 18: Series” and “Lecture 19: Series Convergence Tests, Absolute Convergence”**Link: YouTube: Harvey Mudd College: Professor Francis Su’s*Real Analysis*: “Lecture 18: Series” (YouTube) and “Lecture 19: Series Convergence Tests, Absolute Convergence” (YouTube)Instructions: Please watch these lectures. In the first lecture, Professor Su defines series convergence using partial sums (a.k.a. the Cauchy Criterion). He validates the Comparison Test. He discusses the geometric series. In the second lecture, Professor Su discusses further tests for convergence, along with the definition of absolute convergence. He gives the ratio and root tests. He defines power series. He discusses summation by parts.

Watching these lectures 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.