Instructor: Prof. Madnick (email: jesse.madnick@shu.edu)
Office: McQuaid 213
Lecture Schedule (revised on 8/11)
Textbook: "Real Analysis: A Long-Form Mathematics Textbook (Second Edition)" by Jay Cummings
Lectures: Location TBD
MWF: 11:00 am - 12:15 pm
Office hours: McQuaid 213
Wed: 1:45 pm - 3:00 pm
Thu: 3:15 pm - 5:00 pm
Start here
Calculus review
Lecture Notes
Unit 0: Foundations (prerequisites)
Unit A: The Real Number System
Unit B: Sequences & Series of Numbers
B.1 - B.5: Sequences
B.6 - B.8: Series
Unit C: The Topology of the Real Line
Unit D: Limits & Continuity
Unit E: Differentiation
Unit F: Integration
Unit G: Sequences & Series of Functions
G.1 - G.5: Sequences
G.6 - G.8: Series
Unit H: Metric Spaces
Workshop Problems
Workshop 1 (Wed 9/9) Solutions
Workshop 2 (Wed 9/23) Solutions to #1-6
Workshop 3 (Fri 10/16) Solutions
Workshop 4 (Fri 10/23) Solutions
Workshop 5 (Mon 11/16) Solutions
Workshop 6 (Wed 12/9) Solutions
Tips for exam studying
Memorize all the definitions, propositions, and theorems. Definitions are especially important.
Revisit the PSets and workshop problems: make sure you can solve all of them on your own. (If there's even a single one that you can't solve, you can ask me about it in office hours.)
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 at the end of the lecture notes or from the textbook. (You are welcome to discuss any such problems with me during office hours.)
Ask yourself lots of "what if?" questions to strengthen conceptual understanding. (If you don't know what this is or how to do it, just ask me.)
Tips for reading
"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
Quiz 1 Information
Date: Mon 9/21
Content:
Unit 0 and Unit A. Lectures 1 - 5.
PSets 1 and 2
Workshop 1
Not tested: PSet 0
Structure: Five problems.
State definitions and theorems.
Multiple choice and short response.
Proof. (Unit 0)
Proof. (Unit A)
Proof. (Unit A)
Time management: Solve problems 1 and 2 quickly (~15 minutes total). The idea is to give yourself lots of time (ideally 1 hour) for the three proofs, which take much longer.
Scratch paper will be provided on every quiz. I suggest making good use of the scratch paper to help sketch out ideas and thoughts.
Quiz 2 Information
Date: Wed 10/7
Content:
Sections B.1 - B.5: Lectures 6 - 9.
PSets 3 and 4
Workshop 2
Structure: Five problems. (For problem #5, you get a choice between two problems.)
State definitions and theorems.
True/False.
Proof.
Proof.
Proof. (Choose one from a set of two.)
Time management: Solve problems 1 and 2 quickly (~15 minutes total). The idea is to give yourself lots of time (ideally 1 hour) for the three proofs, which take much longer.
Quiz 3 Information
Date: Wed 10/28
Content:
Sections B.6 - B.7 and C.1, C.3, C.4: Lectures 9 - 15.
PSets 5, 6, 7
Workshop 3
Not tested: Section C.2.
Structure: Six problems. Scoring: Of the four proof problems (#3, #4, #5, #6), only your best three will count. All four proof problems are worth the same amount.
State definitions and theorems.
True/False.
Proof. (Series) (Sec B.6-B.7)
Proof. (Series) (Sec B.6-B.7)
Proof. (Topology) (Unit C) (Choose one from a "topology set" of three)
Proof. (Topology) (Unit C) (Choose another one from a "topology set" of three)
Quiz 4 Information
Date: Wed 11/11
Content:
Unit D. Lectures 15 - 18.
PSets 8 and 9
Workshop 4
Structure: Four problems. All proofs.
One proof.
One proof.
Two proofs.
Two proofs.
Quiz 5 Information
Date: Fri 12/4
Content:
Sections E.1 - E.5 and F.1 - F.3. Lectures 18 - 24.
PSets 10 and 11
Workshop 5
Structure (tentative): Four problems. All proofs.
One proof. (Unit E)
One proof. (Unit E)
One proof. (Units E and F)
One proof. (Units E and F) (Choose one problem from a set of two.)
(Optional: Complete the other problem from the set of two. Your grade for #4 will be whichever one scores higher.)
Final Exam Information
Date: Tue 12/22
Time: 10:10 am - 12:10 pm
Location: Our classroom
Content:
Units E, F, G. Lectures 18 - 30. (Strong emphasis on Unit G.)
PSets 10, 11, 12, 13.
Workshops 5 and 6
Structure (tentative): Seven problems.
True/False. (Units E, F, G)
One proof. (Unit G)
One proof. (Unit G)
One proof. (Unit G)
One proof. (Units F and G)
One proof. (Unit G)
One proof. (Units E and G) (Choose one from a set of three.)