Math 413-01 (Fall 2020)
Real Analysis I, Fall 2020, Math 413 - 01
You will want to check back here regularly as I may post a variety of course materials or links to various resources.
Syllabus is posted to blackboard.
Homework
HW 01 - Due: Friday 9/11 solution
HW 02 - Due: Friday 9/18 solution
HW 03 - Due: Monday 10/05 solution
HW 04 - Due: Friday 10/16 solution
HW 05 - Due: Friday 10/23 solution
HW 06 - Due: Friday 11/06 (moved to Weds 11/11) solution
HW 07 - Due: Monday 11/16 solution
HW 08 - Due: Wednesday 12/2 solution
HW 09 - Due: Friday 12/11 solution
Quizzes
Quiz 01 - Due: Wednesday 10/7 solution
Quiz 02 - Due: Wednesday 10/28 solution
Quiz 03 - Due: Friday 12/4 solution (updated)
Quiz 04 - solution
Exams
Exam 1 solution
Course Notes
These notes are provided as a supplement to what is covered in class and in the textbook. I will update them from time to time. These notes are based in a large part on Dr. Dichone's MATH 413 Spring 2020 course. They are essentially based on the textbook, keeping the definitions and theorems label from the text, but I sometimes modify the language and explain things differently or change the proofs somewhat. I also put in my own example problems and some supplementary notes to cover things that I thought the book wasn't clear enough on. Errata for the textbook can be found here on the author's website.
Chapter 1 and preliminaries:
You will want to check back here regularly as I may post a variety of course materials or links to various resources.
Syllabus is posted to blackboard.
Homework
HW 01 - Due: Friday 9/11 solution
HW 02 - Due: Friday 9/18 solution
HW 03 - Due: Monday 10/05 solution
HW 04 - Due: Friday 10/16 solution
HW 05 - Due: Friday 10/23 solution
HW 06 - Due: Friday 11/06 (moved to Weds 11/11) solution
HW 07 - Due: Monday 11/16 solution
HW 08 - Due: Wednesday 12/2 solution
HW 09 - Due: Friday 12/11 solution
Quizzes
Quiz 01 - Due: Wednesday 10/7 solution
Quiz 02 - Due: Wednesday 10/28 solution
Quiz 03 - Due: Friday 12/4 solution (updated)
Quiz 04 - solution
Exams
Exam 1 solution
Course Notes
These notes are provided as a supplement to what is covered in class and in the textbook. I will update them from time to time. These notes are based in a large part on Dr. Dichone's MATH 413 Spring 2020 course. They are essentially based on the textbook, keeping the definitions and theorems label from the text, but I sometimes modify the language and explain things differently or change the proofs somewhat. I also put in my own example problems and some supplementary notes to cover things that I thought the book wasn't clear enough on. Errata for the textbook can be found here on the author's website.
Chapter 1 and preliminaries:
Supplement: the construction of the natural numbers and integers
Supplement: brief background material on sets, relations, and functions, etc.
Supplement: brief background material on sets, relations, and functions, etc.
1.7 - Real numbers and ordered fields
Supplement: exponentiation/roots of real numbers
1.8 - Some properties of real numbers
Chapter 2: Sequences
Chapter 2: Sequences
2.1 - Convergence
2.2 - Limit theorems
Supplement: general exponentiation of a sequence
2.3 - Infinite limits
2.4 - Monotone sequences
2.5 - Cauchy sequences
2.6 - Subsequences
Supplement: the natural exponential and logarithm
Chapter 3: Limits of functions
3.1 Limits at infinity
3.2 Limit at a real number
3.3 Sided Limits
Chapter 4: Continuity
4.1 - Continuity of a function
4.2 - Discontinuity of a function
4.3 - Properties of continuous functions
4.4 - Uniform continuity
Chapter 5: Differentiation (*I'm still hoping to post some formatted notes for part of chapter 5*)
5.1 - Derivative of a function
5.2 - Properties of derivatives
5.3,4,5 - Mean value theorems, Higher derivatives, L'Hopital's rule
Chapter 6: Integration
We'll cover integration in Math 414 next semester
Chapter 7: Infinite series
- I won't be posting any notes for this chapter
Chapter 8: Sequences and series of functions
8.1,2,3 - Pointwise & Uniform convergence
- I won't post notes for 8.4
Scanning work for turning in:
When turning in work via email, please convert it into a single high resolution pdf document (per assignment). This will make it easiest for me to keep everything organized and to write comments on the work to send back to you.
How to combine pdfs with LaTeX:
Here is an example tex file
Here are the pdfs that are combined with it: pdf01 pdf02 pdf03 printer-test-page
Here is the resulting combined pdf file
It is imperative that you create high quality scans. You can achieve this with a smartphone app and a stable holder for your phone. You can simply balance your phone on a baking rack supported by textbooks or a counter-weighted board balanced on a box. Please make sure the lighting is decent. If you just Google "diy document camera" you will find numerous design choices for setting your phone on a stable base.
CamScanner is an app that I use to scan from phone to pdf. You can take multiple pictures and it will output a single pdf. It's available for both Android and iPhone. I am sure there are many similar apps. This one leaves a watermark with the free version. That's fine.
If you have a local scanner, that should work too. I prefer 300dpi and color, but 150dpi should suffice. Please check your pdf document so that the work is legible. Sometimes small writing or light writing (such as soft pencil) doesn't scan well. I suggest using blue or black ink or a sufficiently dark pencil for your written work. You could even try photos of boardwork, but make sure to organize it into a single document (per assignment).
Course notes:
You are advised to acquire a copy of the textbook. I will provide some of my own notes soemtimes in order to supplement the text.
Course plan:
Here is a weekly outline of the course and material covered. This is a tentative plan, but we will likely not vary from this too much.
Studying and learning advice:
Studying is generally a very personal endeavor. You are advised to figure out what works for you. For most, mathematics is best learned by active engagement. Solving many problems and performing many calculations is advised. Often the best learning occurs when a problem is attempted multiple times with failure before succeeding.
Reflection is a key component to learning. Whether you solved a problem correctly or not or if you do or don't feel like you understand a particular concept, focusing on it with intent and active attention will help you make progress. Solving the same problem multiple times even if you already achieved the correct solution will help hard-wire the concepts and methodology in your brain. Try different approaches. Check for similar examples online or in other course materials. Mathematics Stack Exchange has a great wealth of information as well where many students have probably already asked the same questions you have. There are many other online math forums as well.
Persistence is absolutely necessary! You will be rewarded by your efforts!
2.2 - Limit theorems
Supplement: general exponentiation of a sequence
2.3 - Infinite limits
2.4 - Monotone sequences
2.5 - Cauchy sequences
2.6 - Subsequences
Supplement: the natural exponential and logarithm
Chapter 3: Limits of functions
3.1 Limits at infinity
3.2 Limit at a real number
3.3 Sided Limits
Chapter 4: Continuity
4.1 - Continuity of a function
4.2 - Discontinuity of a function
4.3 - Properties of continuous functions
4.4 - Uniform continuity
Chapter 5: Differentiation (*I'm still hoping to post some formatted notes for part of chapter 5*)
5.1 - Derivative of a function
5.2 - Properties of derivatives
5.3,4,5 - Mean value theorems, Higher derivatives, L'Hopital's rule
Chapter 6: Integration
We'll cover integration in Math 414 next semester
Chapter 7: Infinite series
- I won't be posting any notes for this chapter
Chapter 8: Sequences and series of functions
8.1,2,3 - Pointwise & Uniform convergence
- I won't post notes for 8.4
Scanning work for turning in:
When turning in work via email, please convert it into a single high resolution pdf document (per assignment). This will make it easiest for me to keep everything organized and to write comments on the work to send back to you.
How to combine pdfs with LaTeX:
Here is an example tex file
Here are the pdfs that are combined with it: pdf01 pdf02 pdf03 printer-test-page
Here is the resulting combined pdf file
It is imperative that you create high quality scans. You can achieve this with a smartphone app and a stable holder for your phone. You can simply balance your phone on a baking rack supported by textbooks or a counter-weighted board balanced on a box. Please make sure the lighting is decent. If you just Google "diy document camera" you will find numerous design choices for setting your phone on a stable base.
CamScanner is an app that I use to scan from phone to pdf. You can take multiple pictures and it will output a single pdf. It's available for both Android and iPhone. I am sure there are many similar apps. This one leaves a watermark with the free version. That's fine.
If you have a local scanner, that should work too. I prefer 300dpi and color, but 150dpi should suffice. Please check your pdf document so that the work is legible. Sometimes small writing or light writing (such as soft pencil) doesn't scan well. I suggest using blue or black ink or a sufficiently dark pencil for your written work. You could even try photos of boardwork, but make sure to organize it into a single document (per assignment).
Course notes:
You are advised to acquire a copy of the textbook. I will provide some of my own notes soemtimes in order to supplement the text.
Course plan:
Here is a weekly outline of the course and material covered. This is a tentative plan, but we will likely not vary from this too much.
|
Week |
Dates |
Chapter |
Topics |
|
1 |
8/31 to 9/5 |
1 |
Real
number system |
|
2 |
9/7 to 9/12 |
1 |
Real
number system |
|
3 |
9/14 to 9/19 |
2 |
Sequences |
|
4 |
9/21 to 9/26 |
2 |
Sequences |
|
5 |
9/28 to 10/3 |
2 |
Sequences |
|
6 |
10/5 to 10/10 |
2 |
Sequences |
|
7 |
10/12 to 10/17 |
3 |
Function
limits |
|
8 |
10/19 to 10/24 |
3 |
Function
limits |
|
9 |
10/26 to 10/31 |
4 |
Continuity |
|
10 |
11/2 to 11/7 |
5 |
Differentiation |
|
11 |
11/9 to 11/14 |
5 |
Differentiation |
|
12 |
11/16 to 11/21 |
7 |
Infinite
series |
|
13 |
11/23 to 11/28 |
7 |
Infinite
series |
|
14 |
11/30 to 12/5 |
8 |
Sequences/series
of functions |
|
15 |
12/7 to 12/12 |
8 |
Sequences/series
of functions |
|
16 |
12/14 to 12/19 |
Final |
|
Studying and learning advice:
Studying is generally a very personal endeavor. You are advised to figure out what works for you. For most, mathematics is best learned by active engagement. Solving many problems and performing many calculations is advised. Often the best learning occurs when a problem is attempted multiple times with failure before succeeding.
Reflection is a key component to learning. Whether you solved a problem correctly or not or if you do or don't feel like you understand a particular concept, focusing on it with intent and active attention will help you make progress. Solving the same problem multiple times even if you already achieved the correct solution will help hard-wire the concepts and methodology in your brain. Try different approaches. Check for similar examples online or in other course materials. Mathematics Stack Exchange has a great wealth of information as well where many students have probably already asked the same questions you have. There are many other online math forums as well.
Persistence is absolutely necessary! You will be rewarded by your efforts!