## MATH 4310, Introduction to Real Analysis I (SPRING 2021)

Class Web Page

Course Syllabus

Course Textbook

Welcome to Math 4310, Introduction to Real Analysis I, Spring 2021. Take a look at the course syllabus, and the course textbook.

This web site will be where you find

- course lecture videos
- assignments
- announcements
- handouts and other course material

so check the site regularly.

Class will be over Zoom at 991-699-1928 at the specified times, starting January 19, 2021. The first videos and assignment will be posted soon.

For extra help, we have to LA's working with 4310 students.
Nhat Pham will hold problem sessions every Friday morning from 9:00 - 10:30, and

Alex Semyonov will have office hours on Monday 7:00-8:00pm and Tuesday 2:00-3:00pm.

I hope you all take advantage of our LA's! If you want to talk to me, send me an email and we will arrange a time to meet on zoom.

Take a look at this quick video introduction

### Chapter 2

Here is the first video lecture on section 7 of the text (limits of sequences).
Here is lecture #2 where I prove the "Squeeze Lemma".
Here is lecture #3 on bounded monotonic sequences. (they converge!)
Here is lecture #4 on Limsup and Liminf.
Here is lecture #5 on Cauchy sequences.
As promised, here is lecture #6 on subsequences that will hopefully help it all make sense.
Here is lecture #7 on series and the comparison test.
Here is lecture #8 on the integral test and alternating series test.

Homework problems for Thursday 1/21 and Tuesday 1/26
Section 7, #1,2,4,5

Section 8, #1,2,3,4,10

Note: When doing these problems (and in general) try to think your way through the proof before starting. If you are *positive*
that you know how to write down a proof, then you can skip the problem. If not, start to write and see where you get stuck.

And expect a quiz based on the homework first thing tomorrow (tuesday). This will help me guage how we're com prehending the material so far.

homework for Tuesday Feb 2. We will look at a few from section 9 on thursday January 28.
Section 9, #1,2,4,8,12

Homework for Tuesday Feb 2 and Thursday Feb 4
Section 10, #1,4,6,7,8-11

Section 11, #2,3,4,9,10

Here are some solutions for HW problems from section 10, including quiz#2.

Homework for Tuesday Feb 9 and Thursday Feb 11
Section 11, #2,3,4,9,10 (we didn't get to these last week)

Section 12, #1,2,3,4,14

Check out Video #6 above to help with this weeks lessons.

Homework for Tuesday Feb 16, 18
Section 12, #1,2,3,4,14 (we will discuss on 2/16)

Here are some solutions for some HW problems from Section 12.

Section 14, #1-7 (For 2/18. I'll add some problems for the following week)

Homework for Tuesday Feb 23
Section 15 #1,2,3,4,6 (see video #8)

### Chapter 3 on continuity

Here is lecture #9 on the definition of continuity, with examples.
Here is lecture #10 on uniform continuity.

Homework problems for March 2,4
Section 17 #1,2,3,8 (already done most of these)

Section 17 #9,10,12 (I just realized that one of the examples I did on video #9 is problem #14)

Section 18 #2,4,5,6,7

Homework problems for March 9,11 (Don't forget, I'll post the Midterm exam right after class on Thursday, March 11.)
Section 19 #1,2,3,4,5,6

Here is the midterm exam due March 12 before midnight MDT.
And here are solutions for the exam.
### Chapter 4 on Sequences and Series of Functions

Here is lecture #11 on uniform convergence.
Homework problems for March 18 and 23
Section 23 #1,2,4,7,8,9

Section 24 #1,2,3,4,5,10,11,13

Homework problems for March 25 and 30
Section 25 #3-10

Here are solutions for a couple of the HW problems from Section 25 that I didn't get quite right in class.
Homework problems for April 6
Section 26 #6,7,8 (part 8c needs Abel's theorem)

Here is lecture #12 on Abel's Theorem.
### Chapter 5 on differentiation

Homework problems for April 8 and 13
Section 28 #14,16

Section 29 #1,2,3,5,19,11,14

Read over section 31 on Taylor's Theorem and work the following problems by April 15.

Section 31 #1,2,5,6

### Chapter 6 on Integration

Here is lecture #13 on Riemann integration.
Look over section 32 before class on April 27.

Homework problems for April 29 and May 4
Section 32 #1,2,6,7

Section 33 #3,4,5,10

This will probably be as far as we get this semester!

### Final Exam

Final Exam: The only strong opinions on the timing of the exam were hoping to have the test earlier. This is fine by me,
so let's have the exam right after our review session on Thursday, May 6. The format will be just like the midterm:
I'll post the test right after class and it will be due on Friday, May 7, at midnight MDT. That will give you plenty of time.
The test will be comprehensive, but will stress the material we've covered since the midterm exam.
Here is a study guide for the final exam.

And here is the final exam.
You should have received your graded finals today (May 12). If you have any concerns, let me know as soon as possible.
Here are solutions the exam. As you probably realize, I made this test harder than the midterm
since the class, as a whole, demonstrated that I could ask more chalanging questions. So the numerical grades were lower (the average was about 72/100).
But this does not mean the letter grades were lower. I curved the exam so 80-100 are A's and less than 80 range from B+ to B-.
Everybody learned some analysis this semester. It was a pleasure to try to explain this material to you, and I hope you all enjoyed the course.