MAT337. Introduction to Real Analysis

Fall 2018

Web page:

Class Location & Time: Tue, 1:00PM - 2:00 PM; Thu, 11:00 AM - 1:00 PM; NE2190

Instructor: Ilia Binder (, DH3026.
Office Hours: Tue 2:00 PM - 3:00 PM and Thu 10:00 AM-11:00 AM

Teaching Assistant: Belal Abuelnasr, ( ).
Office Hours:  Fri, 10-11 AM, DH3050.

Textbooks: Understanding Analysis, Second Edition, by Stephen Abbott. This book is provided as a free electronic resource to all UofT students through the library website. Click on the following link to access the textbook (you may be required to enter your UTORid and password):

Prerequisites:  MAT102H5, MAT224H5/MAT240H5, MAT212H5/MAT244H5, MAT232H5/MAT233H5/MAT257Y5
Exclusions:  MAT337H1, MAT357H1,MATB43H3, MATC37H3

Prerequisites will be checked, and students not meeting them will be removed from the course by the end of the second week of classes. If a student believes that s/he does have the necessary background material, and is able to prove it (e.g., has a transfer credit from a different university), then s/he should submit a 'Prerequisite/Corequisite Waiver Request Form'.

The course is the rigorous introduction to Real Analysis. We start with the careful discussion of The Axiom of Completeness and proceed to the study of the basic concepts of limits, continuity, Riemann integrability, and differentiability.

Topics covered in class.

September 6: An introduction. Real numbers and the Axiom of Completeness. Section 1.3.
September 11: The Axiom of Completeness. Nested Interval property. Sections 1.3, 1.4.
September 13: Nested Interval property. Archimedean property. Definitions of the limit of a sequence (including an alternative definition). Limits and algebraic operations. Sections 1.4, 2.2, 2.3.
September 18: Limits and algebraic operations. Limits and order. Squeezed sequence lemma.Section 2.3.
September 20: The Monotone Convergence Theorem. Iterated sequences. Positive series. Liminf and limsup. Section 2.4.
September 25: Liminf and limsup. Subsequences and their limits. Bolzano-Weierstrass Theorem. Section 2.5.
September 27: Bolzano-Weierstrass Theorem. Cauchy Criterion. Series. Sections 2.5, 2.6, 2.7.
October 2: Open and closed sets. Interrior, exterior, and border points. Section 3.2.
October 4: Interrior, exterior, and border points. Compact sets. Heine-Borel Theorem. Sections 3.2, 3.3.
October 16: Heine-Borel Theorem. Baire's Theorem. Sections 3.3, 3.5.
October 18: Functional limits. Sequential criterion. Continuity. Sections 4.2, 4.3.
October 23: Continuity and compact sets. Uniform continuity. Section 4.4.
October 25: Uniform continuity and compact sets. The Intermediate value Theorem. Differentiability (including an alternative definition). Darboux's Theorem. Sections 4.4, 4.5, 5.2.
October 30: Rolle's theorem. The Mean Value Theorem. L'Hospital rule. Pointwise and Uniform convergence. Sections 5.3, 6.2.
November 1: Uniform convergence. Continuity of uniform limit. Uniform convergence and differentiation. Sections 6.2, 6.3.
November 6: Midterm review.
November 8: Midterm.
November 13: Uniform convergence and differentiation. Uniform convergence of series. Sections 6.3, 6.4.
November 15: Power series. Section 6.5.
November 20: Riemann Integration. Section 7.2.
November 22: Riemann Integration: criterion of integrability, non-integrable functions integrability of continuous functions, additivity and algebraic properties of Riemann integral. Sections 7.2, 7.3, 7.4.
November 27: Algebraic properties of Riemann Integral. Integrability of Uniform limit. Section 7.4.
November 29: The Fundamental Theorem of Calculus. Integration by parts. Riemann integrability criterion. Sections 7.5, 8.1.
December 4: Final review.

Homework. The assignments should be submitted through Quercus. To submit, you can scan or take a photo of your work (or write your work electronically). Please make sure that the images are clear and easy to read before you submit them.

Assignment #1, due September 13: The assignment is based on the material you have learned in MAT102.
Please do the following exercises from the textbook: 1.2.3, 1.2.4, 1.2.5, 1.2.7, 1.2.8, 1.2.9, 1.2.10, 1.2.11, 1.2.12, 1.2.13.

Assignment #2, due September 20.

Assignment #3, due October 4.

Assignment #4, due October 18.

Assignment #5, due October 25.

Assignment #6, due November 1.

Assignment #7, due November 8.

Assignment #8, due November 15.

Assignment #9, due November 22.

Assignment #10, due November 29.

Tutorials and presentations. Each student must be registered in one of the tutorials (on ROSI). The attendance of tutorials is mandatory. Based on the homework assignments, the students will be selected to present some of the homework problems at the tutorials. An unexcused absence at the tutorial on the day you are selected for the presentation will result in zero credit for the presentation. 
Tutorials will begin on Friday of the second week of classes. 

Quiz. There will be a one-hour in-tutorial quiz on Friday, September 28, or Monday, October 1, depending on your tutorial section. No aides are allowed for this quiz. The quiz will cover the material of the sections 1.3, 1.4, 2.2, 2.3, 2.4.
Recommended preparation (do not hand in): problems 1.3.2, 1.3.3, 1.3.6, 1.3.8, 1.4.8, 2.2.2, 2.2.4, 2.3.2, 2.3.7, 2.4.1, 2.4.6, 2.4.8.

Midterm Test. There will be a two-hour in-class midterm test on Thursday, November 8. No aides are allowed for this test. The test will cover the material of the sections 1.3, 1.4, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 3.2, 3.3, 3.5, 4.2, 4.3, .4.4, 4.5, 5.2, 5.3.
Recommended preparation: assignment #7, and (do not hand in): all the quiz review problems, 2.5.9, 2.6.4, 2.7.7, 3.2.8, 3.3.8, 3.5.9, 4.2.4, 4.3.6, 4.4.11, 4.5.6, 5.2.10, 5.3.4.

Final exam. The final exam will be held on Wednesday, December 12, 5-8pm, at KN137. No aides are allowed for this test.
The exam will cover the material of the sections 1.3, 1.4, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 3.2, 3.3, 3.5, 4.2, 4.3, .4.4, 4.5, 5.2, 5.3, 6.2, 6.3, 6.4, 6.5, 6.6 (up to Theorem 6.6.2), 7.2, 7.3, 7.4, 7.5, 8.1 (up to Theorem 8.1.2).
You will be required to state and prove in detail one of the following Theorems from the textbook: 2.4.2, 2.5.5, 3.3.4, 4.2.3, 4.3.9, 4.4.1, 4.4.2, 4.4.7, 5.2.7, 5.3.2, 6.2.6, 6.4.4, 7.2.8, 7.5.1.
Recommended preparation (do not hand in): all the quiz and midterm review problems, 6.2.3, 6.2.13, 6.2.14, 6.2.15, 6.3.1, 6.3.6, 6.4.2, 6.4.4, 6.4.10, 6.5.2, 6.5.8, 7.2.3, 7.3.2, 7.3.5, 7.4.3, 7.4.10, 7.5.2, 7.5.4.
Additional office hours: Tuesday, December 11, 12 - 1. Location: DH3000 .

Grading. Grades will be based on the best eight out of ten homework assignements (10%), an in-tutorial quiz (10%), an in-lecture midterm test (25%), tutorial presentations (15%), attendance of tutorials and active participation in the discussions (5%), and Final exam (35%). I will also occasionally assign bonus problems.

Late work. No late work will be accepted. Special consideration for late assignments or missed exams must be submitted via e-mail within a week of the original due date. There will be no make-up quiz, midterm test, or final. Justifiable absences must be declared on ROSI, undocumented absences will result in zero credit.

E-mail policy.
E-mails must originate from a address and contain the course code MAT337 in the subject line. Please include your full name and student number in your e-mail.

Academic Integrity.
Honesty and fairness are fundamental to the University of Toronto’s mission. Plagiarism is a form of academic fraud and is treated
very seriously. The work that you submit must be your own and cannot contain anyone elses work or ideas without proper
attribution. You are expected to read the handout How not to plagiarize ( and to be familiar with the Code of behaviour on academic matters, which is linked from the UTM calendar under the link Codes and policies.