MTH435 
Introduction to Mathematical Analysis I
 
MW 3:00-4:15 PM
Lippitt Hall 205

Instructor    Araceli Bonifant  
Office: Lippitt Hall 200 B
Phone: 874-4394
Email: bonifant@math.uri.edu

Office Hours: Tuesday 2:00PM, Wednesday 2:00PM

Book: Introduction to Real Analysis, 3rd Edition by Robert G. Bartle and Donald R. Sherbert.

Course Description. This course is an introduction to Real Analysis. In this course we will look more deeply at the concepts you learned in your Calculus classes. We will discuss how to prove many of the facts you have used in the past. You will learn precise definitions of notions, get a deeper understanding of concepts, and make your reasoning more rigorous. A very important component of this course is to expose you to proofs. The goal is for you to learn how to write proofs.

Prerequisites. Mth 243 Calculus for Functions of Several Variables, Mth 307 Introduction to Mathematical Rigor is strongly recommended.


Tentative List of Topics:

  • The Real Number System: Algebraic and order properties of ${\mathbb R}$, Absolute value, The completeness Property of ${\mathbb R}$, Applications of the supremum property. Intervals, Infinite sets.

  • Sequences: Sequences and their limits, Subsequences and the Bolzano-Weierstrass theorem, The Cauchy criterion.

  • Limits: Limit theorems.

  • Topology of Real Numbers: Open and close sets in ${\mathbb R}$, compact sets, metric spaces.

  • Continouous Functions: Continuous functions, Uniform continuity, Monotone and Inverse functions.

  • Differentiation: The mean value theorem, L'Hospital's rules, Taylor's theorem.

    • Homework Chapters I, II and III
    Sec 1.1, Page 11: # 3,5,6,8,9,14,16,20
    Sec 1.2, Page 15: # 2, 7, 15
    Sec 1.3, Page 21: # 6, 8, 11, 12
    Sec 2.1, Page 29: # 1, 2, 6, 7, 8, 13, 15, 23, 26
    Sec 2.2, Page 34: # 3, 6, 7, 10, 16
    Sec 2.3, Page 38: # 2, 3, 4, 9
    Sec 2.4, Page 43: # 4, 5, 6, 12, 14
    Sec 2.5, Page 50: # 2, 7, 8, 10, 12
    Sec 3.1, Page 59: 5d, 6c, 7, 11, 15, 16
    Sec 3.2, Page 67: 1, 6, 7, 10, 13, 16, 17 , 18
    Sec.3.3, Page 74: 3, 4, 10, 13 Hand in: 13
    Sec.3.4, Page 80: 1, 3, 7, 8, 12, 14 Hand in: 3
    Sec.3.5, Page 86: 2, 4, 8, 9, 10 Hand in: 4
    Sec.3.6, Page 88: 3, 5, 7, 8, 9 Hand in: 7
    Sec.3.7, Page 95: 1, 2, 3, 12, 14 Hand in: 12
    Sec.4.1, Page104: 1, 9, 11, 14 Hand in: 14
    Sec 4.2, Page110: 2, 8, 13, 14 Hand in: 14
    Sec.5.1, Page124: 3, 4, 9, 10, 11, 13 Hand in: 11
    Sec 5.2, Page128: 1, 4, 5, 12 Hand in: 12
    Sec.5.3, Page129: 4, 6, 9a, 13, 17 Hand in: 13
    Sec.5.4, Page144: 2, 4, 6, 8, 9 Hand in: 8


    Also PROVE
    * If a, b are real numbers then |a-b| \leq |a| + |b|
    * The generalized triangle inequality
    * The supremum is unique
    * Strong Induction = Weak Induction
    * Prove the Principle of Mathematical Induction (second version)
    Note: Some of this proofs are in the book. So essentially I just want you to study them.

    Evaluation Policy:

  • Homeworks and Quizzes             30%
  • Midterm Exam                             30 %    Monday November 2nd.
  • Final Exam                                   40 %    Fri. Dec 18        3:00 pm - 6:00 pm

    • You are expected to abide by the University's civility policy:

    "The University of Rhode Island is committed to developing and actively protecting a class environment in which respect must be shown to everyone in order to facilitate the expression, testing, understanding, and creation of a variety of ideas and opinions. Rude, sarcastic, obscene or disrespectful speech and disruptive behavior have a negative impact on everyone's learning and are considered unacceptable. The course instructor will have disruptive persons removed from the class."

    Cell phones, IPods, beepers and any electronic device must be turned off in class!!

    You are required to do your own work unless specifically told otherwise by your instructor. In support of honest students, those discovered cheating on assignments or exams will receive a grade of zero on the assignment or exam. Use of unauthorized aids such as cheat sheets or information stored in calculator memories, will be considered cheating. The Mathematics Department and the University strongly promote academic integrity.

    Illness Due to Flu: The H1N1 Flu Pandemic may impact classes this semester. If any of us develop flu-like symptoms, we are being advised to stay home until the fever has subsided for 24 hours. So, if you exhibit such symptoms, please do not come to class. Notify me at 874-4394 or email me at bonifant@math.uri.edu of your status, and we will communicate by email or by phone. We will work together to ensure that course instruction and work is completed for the semester.

    The Centers for Disease Control and Prevention have posted simple methods to avoid transmission of illness. These include: covering your mouth and nose with a tissue when coughing or sneezing; frequently washing your hands to protect from germs; avoiding touching your eyes, nose and mouth; and staying home when you are sick. For more information, please view
    http://www.cdc.gov/flu/protect/habits.htm

    URI information on the H1N1 will be posted on the URI website at
    http://www.uri.edu/news/h1n1, with links to the
    http://www.cdc.gov site.

    Special Accommodations: Students who need special accommodations and who have documentation from Disability Services should make arrangements with their instructor as soon as possible. Students should contact Disability Services for Students, Office of Student Life, 330 Memorial Union, 874-2098.