01:640:311 Introduction to Real Analysis I, Sections T6, Summer 2016

Office hours: Tuesdays and Thursdays 5:00PM - 6:00PM, 8:30PM - 9:00PM, Hill 624 or by appointment.

Email: cl.volkov at rutgers dot edu (for friends) / fq15 at scarletmail.rutgers.edu (for teaching)

In the Summer of 2016 I taught 640:311 (Introduction to Real Analysis I) for Section T6.
Here are the Syallbus and the Schedule.
All announcements and workshop assignments are made on sakai only. Please check the announcement tab and the email archive tab to make sure you don't miss anything.
For 311 students, I have two requirements

  • Please make sure you have a solid understanding on the math 300 class (Introduction to Mathematical Reasoning). You can review the knowledge using the following material
    Dr. Sussmann's notes on Math 300, Lecture 2, 3 and 4
    This set of notes summarizes the most essential knowledge in that class. On his course website you'll find more related material for reviewing.

  • Please recall the knowledge of Calculus I, especially the graphs of the most commonly seen elementary functions. You can check the following file to recall the knowledge:
    Table of Common Graphs
    Although the main focus is to formulate rigorous argument, in many cases this process is facilitated by the intuition from the graphs.
    Also I'll assume a solid basis of computational skills for this class. Please try problems in Chapter 1 and 2 of famous Russian book
    3193 Problems in Mathematical Analysis
    to test your skills.

Lecture Materials:

  • Lecture 1 (May 31, 2016): Lecture Notes, Homework (due June 2nd): 1.2.1, 1.2.5, 1.2.6, 1.2.7.
    None of the homework problems today needs any kind of induction! Please make sure you don't use it.
    For workshop submissions, Overleaf is a very nice online LaTeX editor that you can use. You can use the template to write your solutions. A LaTeX introduction and an Online Tutorial to LaTeX are provided here to help.
    In case your book hasn't arrived yet, here is the scan of the first few sections.

  • Lecture 2 (Jun. 2, 2016): Lecture Notes, Homework (due Jun. 7th): 1.2.10, 1.2.13, 1.3.3, 1.3.5.
    I have created a Discussion Page on Overleaf, collecting interesting problems I was asked via email. Please check here from time to time.
    Dr. Chris Woodward has agreed to share his lecture notes on math 300. Please find it on sakai.

  • Lecture 3 (Jun. 7, 2016): Lecture Notes, Homework (due Jun. 9th): 1.3.1, 1.3.11, 1.4.2, 1.4.8
    Someone asked about the Well-Ordering Principles and how to prove it. Dr. Sussmann provided a proof here

  • Lecture 4 (Jun. 9, 2016): Lecture Notes, Homework (due Jun. 14th): 1.4.6, 1.4.7, 1.5.2, 1.5.5
    In order to prepare you better for the future classes regarding countability, I'll go over this set of notes in the next lecture, but mostly by handwaving.

  • Lecture 5 (Jun. 14, 2016): Lecture Notes, Homework (due Jun. 16th): 2.2.2, 2.2.3, 2.2.4

  • Lecture 6 (Jun. 16, 2016): Lecture Notes, Homework (due Jun. 21st): 2.3.1, 2.3.2, 2.3.4, 2.3.5

  • Lecture 7 (Jun. 21, 2016): Lecture Notes, Homework (due Jun. 23rd): 2.3.7, 2.3.12, 2.4.1, 2.4.2

  • Lecture 8 (Jun. 23, 2016): Lecture Notes, Homework (due Jun. 28th): 2.5.1, 2.5.2, 2.5.6, 2.5.7
    Some hints to 2.5.1d is provided in the lecture notes. Hopefully it helps.

  • Lecture 9 (Jun. 28, 2016): Lecture Notes, Homework (due Jun. 30th): 2.5.5, 2.6.2, 2.6.3, 2.6.4
    In this set of notes I added the simplification to the arguments to Problem 2.5.2d. My argument for 2.5.2c refuses to be simplified. And it is useful for the homework problem 2.5.5. Please study them well before you attempt 2.5.5.
    The purpose of including series is to show an example on how Cauchy's criterion is applied. Infinite series won't be the main topic for the exam.

  • Lecture 10 (Jun. 30, 2016): Lecture Notes. No homework today. Attempt all other problems in the book and prepare for the coming midterm.
    Here is the Collection of Workshop Solutions so far. In case you are stuck, please find help here.
    The detailed proofs of theorems concerning limit superior and limit inferior is available in my old workshop notes.

  • Lecture 11 (Jul. 5, 2016): Midterm 1, Solutions
    People not doing well in Midterm 1 are welcomed to attend the Second Chance Club. Please find the details here

  • Lecture 12 (Jul. 7, 2016): Lecture Notes, Homework (Due Jul. 12th): 3.2.1, 3.2.2 (excluding d), 3.2.4 (excluding e)

    • I stopped updating this website due the server migration process (that messed a lot of things up). Everything was put on sakai. In order to prepare the future semester better, I put something back here.

  • Lecture 13 (Jul. 12, 2016): Lecture Notes, Homework (Due Jul. 14th): 3.2.4, 3.2.8, 3.2.14, 3.3.1, 3.3.5.

  • Lecture 14 (Jul. 14, 2016): Lecture Notes, Homework (Due Jul. 19th): 3.3.4, 3.3.6, 3.3.9, 3.3.11.

  • Lecture 15 (Jul. 19, 2016): Lecture Notes, Homework (Due Jul. 21st): 4.2.1(a), 4.2.2, 4.2.4, 4.2.5, 3.3.2.

  • Lecture 16 (Jul. 21, 2016): Lecture Notes, Homework (Due Jul. 26th): 4.2.3, 4.2.6, 4.2.7, 4.3.1. Also if you got 3.3.11 wrong, please resubmit it.

  • Lecture 17 (Jul. 26, 2016): Lecture Notes, Homework (Due Jul. 28th): 4.3.6, 4.3.8, 4.3.9, 4.4.2.
    Comments to other problems: 4.3.2 - 4.3.8 are very good exercises for you to get acquainted to the knowledge. 4.3.9 - 4.3.12 are important facts in the theory. 4.3.13 and 4.3.14 are fun but not so essential.

  • Lecture 18 (Jul. 28, 2016): Lecture Notes, Homework (Due Aug. 2nd): 4.4.2, 4.4.3, 4.4.4, 4.4.6, 4.5.2 (skip (e)), 4.5.7.

  • Lecture 19 (Aug. 2, 2016): Midterm 2, Solutions

  • Lecture 20 (Aug. 4, 2016): Lecture Notes, Homework (Due Aug. 9th): 5.2.2, 5.2.5, 5.2.7, 5.2.9.

  • Lecture 21 (Aug. 9, 2016): Lecture Notes, Homework (Due Aug. 11th): 5.3.2, 5.3.4, 5.3.6, 5.3.7.

  • Lecture 22 (Aug. 11, 2016): Lecture Notes. No more homework.

  • Lecture 23 (Aug. 16, 2016): Final Exam.

Supplementary Reading