Real Analysis I

MATH 324 : Spring 2022

Department of Mathematics - SUNY Geneseo

Course Description

A study of the underlying theory of elementary calculus. Topics include the structure and properties of the real numbers, sequences, functions, limits, continuity, the derivative, the Riemann integral, and Taylor's theorems.

Instructor

Cesar Aguilar, South Hall 325A

Office Hours

Tue 8:30-9:30, 10:45-11:45
Thu 10:45-11:45

Class Meetings

MWF 1:30-2:20 pm Fraser 108

Software

  1. Intro to LaTeX Video
  2. Overleaf
  3. LaTeX Tutorial
  4. LaTeX Homework Template
  5. You can also install LaTeX on your machine:
    1. For Mac: MacTeX, MiKTeX
    2. For Windows: MiKTeX

Final Exam

May 18, 2022, 12:00-2:30 pm

Textbook and Resources

  1. An Introduction to Real Analysis by Cesar Aguilar
  2. Introduction to Real Analysis by Bartle and Sherbert, 4th edition or higher

Student File Upload

  Upload your homework, test, lab

Latest

The current week content will be displayed here during the semester. For now, see the Schedule tab.

Homework

Title Due Date Week No.
Homework 1 - Proofs, Induction Feb 2, 2022 1
Homework 2 - Functions, Countability Feb 11, 2022 2
Homework 3 - Real Numbers Feb 21, 2022 3
Homework 4 - Completeness Feb 28, 2022 4
Homework 5 - Sequences Mar 7, 2022 5
Homework 6 - Limit Theorems Mar 21, 2022 7
Homework 7 - Monotone Sequences Mar 29, 2022 8
Homework 8 - Bolzano-Weierstrass Apr 6, 2022 9
Homework 9 - Cauchy Sequences Apr 11, 2022 10
Homework 10 - Series Apr 19, 2022 11
Homework 11 - Limits Apr 26, 2022 12
Homework 12 - Continuity May 2, 2022 13
Homework 13 - Continuity on [a, b] May 9, 2022 14

Schedule

Week 1  Jan 24 - Jan 28
Proofs, Induction
Topics: Proofs, Induction
What to Read: 1.1-1.2
README: Writing Math
Jan 26 First day of classes
HOMEWORK HW 1DUE: Feb 02
Week 2  Jan 31 - Feb 04
Functions, Countability
Topics: Functions, Countability
What to Read: 1.3-1.4
HW DUE HW 1DUE: Feb 02
HOMEWORK HW 2DUE: Feb 11
Week 3  Feb 07 - Feb 11
Real Numbers
Topics: Real Numbers
What to Read: 2.1-2.3
HW DUE HW 2DUE: Feb 11
HOMEWORK HW 3DUE: Feb 21
Week 4  Feb 14 - Feb 18
Completeness Axiom, Applications
Topics: Completeness Axiom, Applications
What to Read: 2.4-2.5
Feb 16 Diversity Summit - No classes
HOMEWORK HW 4DUE: Feb 28
Week 5  Feb 21 - Feb 25
Sequences and Their Limits
Topics: Sequences and Their Limits
What to Read: 3.1
Test 1 Review: Review Questions
HW DUE HW 3DUE: Feb 21
HOMEWORK HW 5DUE: Mar 07
NEXT WEEK Test 1 on Mar 2, 1:30 PM – 2:30 PM
Week 6  Feb 28 - Mar 04
Sequence Limit Theorems
Topics: Sequence Limit Theorems
What to Read: 3.2
HW DUE HW 4DUE: Feb 28
TEST 1 Mar 2, 1:30 PM – 2:30 PM
Week 7  Mar 07 - Mar 11
Monotone Convergence
Topics: Monotone Convergence
What to Read: 3.3
HW DUE HW 5DUE: Mar 07
HOMEWORK HW 6DUE: Mar 21
Week 8  Mar 14 - Mar 18
March Break
Topics: March Break
What to Read: N/A
Mar 14-18 Spring Break - No Classes
HOMEWORK HW 7DUE: Mar 29
Week 9  Mar 21 - Mar 25
Bolzano-Weierstrass Thm.
Topics: Bolzano-Weierstrass Thm.
What to Read: 3.4
HW DUE HW 6DUE: Mar 21
HOMEWORK HW 8DUE: Apr 06
Week 10  Mar 28 - Apr 01
Cauchy Sequences
Topics: Cauchy Sequences
What to Read: 3.6
HW DUE HW 7DUE: Mar 29
HOMEWORK HW 9DUE: Apr 11
Week 11  Apr 04 - Apr 08
Series
Topics: Series
What to Read: 3.7
Test 2 Review: Sequences, Series
HW DUE HW 8DUE: Apr 06
HOMEWORK HW 10DUE: Apr 19
NEXT WEEK Test 2 on Apr 13, 1:30 PM – 2:20 PM
Week 12  Apr 11 - Apr 15
Limits of Functions
Topics: Limits of Functions
What to Read: 4.1-4.2
HW DUE HW 9DUE: Apr 11
HOMEWORK HW 11DUE: Apr 26
TEST 2 Apr 13, 1:30 PM – 2:20 PM
Week 13  Apr 18 - Apr 22
Continuity
Topics: Continuity
What to Read: 5.1-5.2
Apr 21 GREAT Day - No Classes
HW DUE HW 10DUE: Apr 19
HOMEWORK HW 12DUE: May 02
Week 14  Apr 25 - Apr 29
Continuity on [a,b]
Topics: Continuity on [a,b]
What to Read: 5.3-5.4
HW DUE HW 11DUE: Apr 26
HOMEWORK HW 13DUE: May 09
Week 15  May 02 - May 06
Differentiation, MVT
Topics: Differentiation, MVT
What to Read: 6.1-6.2
Test 3 Review: Test 3 Review
HW DUE HW 12DUE: May 02
NEXT WEEK Test 3 on May 11, 1:30 PM – 2:20 PM
Week 16  May 09 - May 13
Riemann Integral
Topics: Riemann Integral
What to Read: 7.1-7.3
May 12 Last day of classes
HW DUE HW 13DUE: May 09
TEST 3 May 11, 1:30 PM – 2:20 PM

Syllabus

Learning Outcomes

Upon successful completion of MATH 324 - Real Analysis I, a student will be able to:

    Grading Scheme

    Below is the tentative course grading scheme. The grading scheme may change during the semester at the discretion of the instructor. Any changes to the grading scheme will be announced in class before the final exam. If homework assignments are done in groups, then a student must achieve a passing grade in all individual assessments (e.g., tests and final exam) to pass the course.

    ItemPercentage
    Homework25
    Writing Assignment5
    Tests30
    Final40
    GradePercentage
    A94-100
    A−90-93
    B+87-89
    B83-86
    B−80-82
    C+77-79
    C73-76
    C−70-72
    D60-69
    E< 60

    Tests and Exam

    There will be 3-4 tests scheduled evenly throughout the semester. The final exam is scheduled for May 18, 2022, 12:00-2:30 pm. The final exam will be cumulative, that is, any topic covered in the course could be tested in the final exam. There will be no make-up for a missed test or final exam under any circumstances. If a student misses a test and can present evidence of an extenuating circumstance then the weight of the missed test will be redistributed to the final exam weight. Having the cold or flu is not an extenuating circumstance. Examples of extenuating circumstances include a medical emergency, a serious prolonged illness, or the death of a member of your immediate family.

    Homework

    There will be approximately one homework assignment per week. Students will be given approximately one week to complete a homework assignment. All homework assignments must be written in LaTeX (not Word). Instructions will be given on the first day of class on how to obtain and use the LaTeX program. The quickest way to get started with LaTeX is to use the online application called Overleaf and by reading the Learn LaTeX in 30 Minutes tutorial. If homework assignments are done in groups, then a student must achieve a passing grade in all individual assessments (e.g., tests and final exam) to pass the course. I encourage you to collaborate with your colleagues on your assignments/labs but your final submitted work should be your own (see Academic Dishonesty statement below).

    Technology

    Calculators are not permitted during the tests or final exam. However, I encourage you to use your calculator, Maple and other math software, WolframAlpha, and other forms of technology as you study and do your assignments. Both Maple and Mathematica can be downloaded from SUNY Geneseo Software page.

    Office Hours and Math Learning Center

    I encourage you to come to my office (South Hall 325A) whenever you are having trouble with any part of the course material, seeking academic advice, or you just want to chat about mathematics in general. If you want to meet with me outside of my office hours, you will need to make an appointment, preferably via email. I also encourage you to visit the Math Learning Center located in South Hall 332 where you can receive free tutoring on a walk-in basis by highly qualified upper level students. Access to in-person office hours and to the MLC will depend on social distancing guidelines set by the College.

    Email Communication

    I will do my best to reply to student email regarding the logistics of the course within 24 hours during the working week (Mon-Fri). However, due to the potential large volume of emails, inquiries regarding homework problems and/or specific course content should be made during office hours or after class.

    Academic Dishonesty and Plagiarism

    Please read, and follow, Geneseo's Academic Dishonesty and Plagiarism policy. Below is the definition of plagiarism and its consequences as described in SUNY Geneseo's Academic Dishonesty and Plagiarism statement:

    Plagiarism is the representation of someone else's words or ideas as one's own, or the arrangement of someone else's material(s) as one's own. Such misrepresentation may be sufficient grounds for a student's receiving a grade of E for the paper or presentation involved or may result in an E being assigned as the final grade for the course.

    If there is sufficient evidence of academic dishonesty on a homework assignment, all students involved will receive a zero score on the homework assignment and I will provide the department chairperson, the dean of academic planning and advising, and the student(s) with a written report of the violation, the penalty imposed and the counseling provided to the students involved. A second instance of academic dishonesty on a homework assignment will result in a final grade of E for the course for all students involved. Academic dishonesty on a test/final exam will result in a final grade of E for the course.

    Academic Accommodations

    SUNY Geneseo is dedicated to providing an equitable and inclusive educational experience for all students. The Office of Accessibility (OAS) will coordinate reasonable accommodations for persons with disabilities to ensure equal access to academic programs, activities, and services at Geneseo.

    Students with approved accommodations may submit a semester request to renew their academic accommodations. Please visit the OAS website for information on the process for requesting academic accommodations.

    Questions? Contact the OAS by email, phone, or in-person:

    Office of Accessibility Services
    Erwin Hall 22
    585-245-5112
    access@geneseo.edu