Fall 2015 MAT 319: Foundations of Analysis Fall 2015 MAT 320: Introduction to Analysis Schedule TuTh 10:00-11:20 Library E4310 TuTh 10:00-11:20 Math P-131 (through 10/1: joint lectures in Math P-131) Instructor Oleg Viro Xiuxiong Chen Office hours TuTh 11:30-12:30 in Math 5-110 Recitation MW 11:00-11:53 Library E4310 MW 11:00-11:53 Library E4330 TA Michael Albanese Silvia Ghinassi Office hours MW 2:30-3:30 in MLC MW 2:30-3:30 in MLC Description A careful study of the theory underlying topics in one-variable 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. 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, required for math majors. 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 Kenneth Ross Elementary Analysis: The Theory of Calculus, 2nd edition Homework Weekly problem sets will be assigned, and collected in Wednesday recitation. The emphasis of the course is on writing proofs, so please try to write legibly and explain your reasoning clearly and fully. You are encouraged to discuss the homework problems with others, but your write-up must be your own work. 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: 10%, Quizzes: 10%, Midterm I: 20%, Midterm II: 20%, Final: 40%.

Syllabus/schedule (subject to change)
All joint lectures through 10/1 meet in Library E4310.
First recitation on Mon 8/24.

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 8/25 1. Joint class: Introduction, motivation: what are real numbers? Read pages 1-19 Thu 8/27 2. Joint class: Properties of numbers; induction; concept of a field. Tue 9/1 3. Joint class: Completeness axiom for real numbers; Archimedean property. Read pages 20-27 Thu 9/3 4. Joint class: Infinity, unboundedness. Intro to sequences. Read pages 28-38 Tue 9/8 Tue 9/8 No class: day after Labor Day Thu 9/10 5. Joint class: Limit of a sequence. Tue 9/15 6. Joint class: Limit laws for sequences. Read pages 39-55 Thu 9/17 7. Joint class: Divergence to infinity, more formal proofs. Tue 9/22 8. Joint class: Monotone and Cauchy sequences. Read pages 56-65 Thu 9/24 9. Joint class: Subsequences. Tue 9/29 Joint Midterm I in Library E4310. Thu 10/1 10. Joint class: Subsequences.

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