Spring 2023 MAT 319: Foundations of Analysis  Spring 2023 MAT 320: Introduction to Analysis  
Schedule  TuTh 11:30am12:50pm Library W4550 (through 3/2: joint lectures in Library W4550)  TuTh 11:30am12:50pm Frey 305 (through 3/2: joint lectures in Library W4550) 
Instructor  James Waterman  Jason Starr 
Office hours  JW's web card  JS's web card 
Recitation  MW 2:403:35pm Physics P117, Earth & Space 183, Frey 224  MW 2:403:35pm Physics P117 
TA  Alex Rodriguez  
Description  A careful study of the theory underlying topics in onevariable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series.  A careful study of the theory underlying calculus. The real number system. Basic properties of functions of one real variable. Differentiation, integration, and the inverse theorem. Infinite sequences of functions and uniform convergence. Infinite series. Metric spaces and compactness. 
Overview  The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements.  An introductory course in analysis, it provides a closer and more rigorous look at material which most students encountered on an informal level during their first two semesters of Calculus. Students learn how to write proofs. Students (especially those thinking of going to graduate school) should take this as early as possible. 
Prerequisites 
C or higher in MAT 200 or permission of instructor; C or higher in one of the
following: MAT 203, 205, 211, 307, AMS 261, or A or higher in MAT 127, 132, 142,
or AMS 161. Math majors are required to take either MAT 319 or MAT 320  
Textbook  Bartle and Sherbert Introduction to Real Analysis, 4th edition  
Homework  Weekly problem sets will be assigned, and collected in the Thursday lecture. Write on the top of each homework: Your recitation number and your
full name as it appears in the grading center of Brightspace: Last Name, First Name. Also list all sources and people you consulted in preparing your answers (you need not list the textbook, lecturers, or TAs).
The emphasis of this course is on writing proofs, so please
write legibly and explain your reasoning clearly and fully with complete sentences. Your writeup must be your own work, in your own words, based on your own understanding.
Late homework will never be accepted, but under documented extenuating circumstances the grade may be dropped. Your lowest homework grade will be dropped at the end of the class.  
Grading  Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. 
Syllabus/schedule (subject to change)
All joint lectures through 3/2 meet in Library W4550.
The first recitation with new material is on Wed 1/25.
Recommendations on choosing MAT 319 vs MAT 320 will be made based upon your performance on the first midterm and homework to that date.
Tue 1/24  1.  Joint class: Sets, induction (Starr)  Read Sections 1.11.3.  
Thu 1/26  2.  Joint class: Infinite sets. (Waterman)  HW due 2/2: p.10 5, 6, p.15 2, 9, p.22 4  
Tue 1/31  3.  Joint class: Algebraic properties of the real numbers. (Starr)  Read Sections 2.12.3  
Thu 2/2  4.  Joint class: Completeness of the real numbers. (Waterman)  HW due 2/9: p.30 8, 15, p.35 3, 16, p.39 6  
Tue 2/7  5.  Joint class: Applications of the supremum property (Starr)  Read Sections 2.42.5  
Thu 2/9  6.  Joint class: Intervals. (Waterman)  HW due 2/16: p.45 2, 4, 7, p.52 3, 6  
Tue 2/14  7.  Joint class: Sequences and limits. (Starr)  Read Sections 3.13.2  
Thu 2/16  8.  Joint class: Limit theorems. (Waterman)  HW due 3/2: p.61 6, 8, 9, p. 69 6, 16  
Tue 2/21  9.  Joint class: Monotone sequences. (Starr)  Read Sections 3.33.4  
Thu 2/23  Joint Midterm I in Library W4550.  
Tue 2/28  10.  Joint class: Subsequences and the BolzanoWeierstrass Theorem (Starr)  Read Section 3.4  
Thu 3/2  11.  Joint class: Cauchy's criterion (Waterman)  Read Section 3.5. HW due 3/9: p.77 1, 7, p.84 1, p.91 5, 8 
Tue 3/7  12.  Divergent sequences. Infinite series  Read Sections 3.6, 3.7,4.1. 
Thu 3/9  13.  Limits of functions  HW due 3/23:

Tue 3/14  Spring Break  
Thu 3/16  Spring Break  
Tue 3/21  14.  Limits theorems  Read Sections 4.2, 5,1, 5.2. 
Thu 3/23  15.  Continuous functions  HW due 3/30:

Tue 3/28  16.  Uniform continuity  Read Sections 5.3, 5.4. 6.1. 
Thu 3/31  17.  Derivatives  HW due 4/13

Tue 4/5  18.  Mean value theorem and L'Hospital  Read Sections 6.2, 6.3. 
Thu 4/6  Midterm II in class  
Tue 4/11  19.  Taylor's Theorem  Read Sections 6.4, 7.1, 7.2. 
Thu 4/13  20.  Riemann integral  HW due 4/20:

Tue 4/18  21.  The fundamental theorem  Read Sections 7.3, 8.1, 8.2. 
Thu 4/20  22.  Uniform convergence  HW duw 4/27:

Tue 4/25  23.  Absolute convergence  Read Sections 9.1, 9.2, 9.3, 9.4. 
Thu 4/27  24.  Series of functions  HW due 5/4:

Tue 5/2  25.  Open, closed and compact sets  Read Sections 11.1, 11.2, 11.3, 11.4 
Thu 5/4  26.  Metric spaces  HW (not due, but do it):

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