Instructor: Prof. Madnick (email: jesse.madnick@shu.edu)
Office: McQuaid Hall 213
Syllabus (amended 9/27)
Lecture schedule
Textbook: "Real Analysis: A Long-Form Mathematics Textbook" (by Jay Cummings)
Sequel course: Real Analysis 2
(Spring 2025) (MW: 3:30 - 4:45)
Lectures: Corrigan Hall 64
MWF: 11:00 - 12:15
Office hours: McQuaid Hall 213
Workshop problems (optional)
Workshop 1. Solutions to #1, 2, 3, 6.
Workshop 2. Solutions to #1-4, A1-A3, A5.
Workshop 3. Solutions to #1-8.
Workshop 4. Solutions to #1-8.
Workshop 5. Solutions.
Start Here
Style guide for proof writing (by Dana Ernst, adapted from a book of Anders Hendrickson)
Review: Dr. Madnick's Calculus Course Notes
"Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis?" -- Paul Halmos
Tips for exam studying:
Memorize all the definitions, propositions, and theorems.
Revisit the PSets and practice problems: make sure you can solve all of them.
For each theorem we learned: ask whether the converse is true, or whether all the hypotheses are necessary. In this regard, you should know simple counter-examples.
Solve problems from the textbook. (You are welcome to discuss any such problems with me during office hours.)
Strengthen conceptual understanding by asking yourself lots of "what if?" questions. (If you don't know what this means or how to do it, ask me.)
Quiz Information
Date: Mon 9/16
Time & Location: Our classroom during lecture
Content
Unit 0 and Unit A
PSets 1 and 2 and Workshop 1
Structure: Six problems. Max score: 44/44
[8 pts] #1. State definitions.
[6 pts] #2. State theorems & provide counter-examples. (Unit A)
[10 pts] #3. True/False & short proof. (Sec A.4)
[6 pts] #4. One proof. (Unit 0)
[6 pts] #5. One proof. (Sec A.1-A.3)
[8 pts] #6. One proof. (Sec A.1-A.3)
Time management (important): Solve problems #1 and #2 quickly. You'll want to have as much time as possible for problems #3, 4, 5, 6, which take much longer.
Midterm 1 Information
Date: Wed 10/2
Time & Location: Our classroom during lecture
Content
Sections B.1 -- B.5
PSets 3 and 4
Workshop 2
Structure: Six problems. Max score: 37/37
[12 pts] #1. State definitions and theorems. (Sec B.1-B.5)
[7 pts] #2. True/False. (Sec B.1-B.5)
[6 pts] #3. One proof. (Sec B.1-B.3)
[6 pts] #4. One proof. (Sec B.1-B.3)
[6 pts] #5. One proof. (Sec B.1-B.3)
[6 pts] #6. One proof. (Sec B.5)
Scoring: Only your best three solutions among {#3, #4, #5, #6} will count.
Time management (important): Solve problem #1 quickly. You'll want to have as much time as possible for the other problems.
Midterm 2 Information
Date: Wed 10/23
Time & Location: Our classroom during lecture
Content
Sec B.6-B.7 and Sec C.1, C.3, C.4. (Don't worry about Sec C.2.)
PSets 5 and 6A and also the first two problems of PSet 6B.
Workshop 3
The following topics will not be tested: The Cauchy Condensation Test; the Cantor set; Separated sets; Connected sets.
Structure: Five problems. Max score: 40/34
[8 pts] #1. State definitions. (Sec C.1 and C.3)
[8 pts] #2. True/False. (Sec B.6, B.7 and C.1, C.3)
[6 pts] #3. One proof about series. (Choose 1 problem on a "series list" of 3.) (Sec B.6-B.7)
[6 pts] #4. One proof about topology. (Choose 1 problem on a "topology list" of 4.) (Unit C)
[6 pts] #5. One proof about topology. (Choose 1 problem on a "topology list" of 4.) (Unit C)
[+6] Extra Credit. Choose another problem from either list and solve it.
Time management (important): Solve problem #1 quickly. You'll want to have as much time as possible for the other problems.
Midterm 3 Information
Date: Wed 11/13
Time & Location: Our classroom during lecture
Content
Unit D and Sec E.1-E.4
PSets 7, 8, 9. Also: The last two problems on PSet 6B. Also: The first two problems of PSet 10.
Workshop 4
Note: Sections E.5, E.6, E.7 will not be tested on Midterm 3.
For Midterm 3: You may bring 3 pages (each double-sided) of handwritten notes to the exam.
Structure: Five problems. Max score: 28/28
#1. [7 pts] Given a function: Prove it is differentiable / not differentiable. (Sec E.1)
#2. [7 pts] Given a function: Prove it is continuous / discontinuous. (Sec D.1-D.2)
#3. [7 pts] One proof: Uniform continuity. (Sec D.5)
#4. [7 pts] One proof. (Choose one problem on a "Unit E list" of 2.) (Sec E.1-E.4)
#5. [7 pts] One proof. (Choose one problem on a "Unit D list" of 2.) (Sec D.1-D.2 and D.4)
Scoring: Only your best four solutions will count.
Final Exam Information
Date: Tue 12/17
Time: 10:10 am - 12:10 pm
Location: TBA
Content
Units F and G. (Emphasis is on Unit G.) (update: The extra credit problem will be about Unit E.)
PSets 10, 11, 12.
For the final exam: You may bring 3 pages (each double-sided) of handwritten notes to the exam.
Structure: Five problems. Max score: 36/32
#1. [8 pts] Sequences of functions. (Choose one problem on a list of 2.) (Sec G.1-G.3)
#2. [8 pts] Series of functions (Sec G.6)
#3. [8 pts] Sequences of functions. (Sec G.1-G.3)
#4. [8 pts] Integration. (Choose one problem on a "Unit F" list of 2.) (Unit F)
#5. [8 pts] Choose one problem from a "grab bag" list of 4. (Unit F and Sec G.4, G.7, G.8)
E.C. [+4 pts] Unit E problem.
Scoring: Among #1-5: Only your best four solutions will count.