Math 104, Introduction to Real Analysis

Spring 2014

TuTh 3:30-5 PM, 6 Evans

Welcome to the course webpage for Math 104 (Section 3). Here you will find some general info about the course. This is also the place to look for homework assignments, occasional course notes, and other things that might interest you. This webpage is also available through our course page on bspace.

Course Syllabus

The textbook we'll be using is Ross, Elementary Analysis (2nd edition). There are many other books you can look at for reference, for example the classic book by Rudin, Principles of Mathematical Analysis , or Lebl's book which you can download here.

The official prerequisites for this course are MAT 53 and 54. While we won't use too much of what you may have learned in these courses, the assumption is that taking them will have equipped you with the appropriate mathematical maturity needed for this class.

Your grade for the course is determined as follows: 15% for homework, 25% for each of two midterms, 55% for the final, and -20% for your lowest exam score. There will be no make up exams.

Homework (along with occasional supplementary notes) will be posted here and will be due every Tuesday at 5PM in class, or in my office by the same day/time (if you choose the office option, you can slide it under the door). The lowest two homework scores will be dropped. Therefore the policy is that no late homework will be accepted, especially since we will sometimes discuss the solutions to the homework problems in class.

The midterms will be in class on Tuesday, February 25 and Tuesday, April 8.

Real Analysis, or the study of functions over the real numbers, is essential to not only the mathematician, but to the physicist, to the engineer, and beyond. In this course we will learn about the convergence of sequences and series, and will cover rigorously notions of continuity and differentiability of functions over the real numbers. We will also venture into topics such as integration and metric spaces. Much of this has been introduced to you in courses such as Math 1A and 1B but we will focus more on mathematical rigor than on computation, which is key to truly understanding and appreciating the subject.

For many students this course may be the first (or one of the first) course in which they are challenged to think like a mathematician: through homeworks, exams, and hopefully discussions with classmates, students will become comfortable with creating and writing rigorous mathematical proofs. This skill is essential to all upper division courses in mathematics, and to any mathematician in general.

• Preliminaries: some basic set theory; natural, rational, and real numbers; the notion of infinity.
• Sequences and series and convergence properties.
• Continuity, limits, and differentiation.
• Riemann integration and the fundamental theorem of calculus.

• 1/21: Natural numbers, rational numbers, and ordered fields. Sections 1, 2, and 3 in the book.
• 1/23: Construction of the reals via Dedekind cuts, completeness of the real numbers. Sections 4 and 6 in the book.
• 1/28: Completeness of the real numbers, beginning sequences. Sections 4 and 7 in the book.
• 1/30: Limits of sequences, divergent vs. convergent sequences. Sections 7,8,9 in the book.
• 2/4: Monotone sequences, subsequences, Bolzano-Weierstass Theorem. Sections 10 (minus limsup/liminf and Cauchy sequences) and 11 in the book.
• 2/6: Subsequences continued, limit points. Section 11 in the book.
• 2/11: Subsequential limits (aka limit points), Cauchy sequences, starting limsup/liminf. Sections 10 and 11 in the book.
• 2/13: Limsup/liminf, possibly starting series. Sections 12 and 14 in the book.
• 2/18: More on limsup and liminf, series. Sections 12 and 14 in the book.
• 2/20: Series. Section 14 in the book.
• 2/25: MIDTERM 1 in 160 Dwinelle. The mean was 37/50. Solutions are available in Resources on bspace.
• 2/27: Series continued, starting to talk about metric spaces. Sections 15 and 13 in the book.
• 3/4: Metric spaces: sequences, completeness, and beginning topology. Section 13 in the book, and parts of Rudin posted on bspace.
• 3/6: Continuing point set topology and moving on to compactness and Heine-Borel. Section 13 in the book, and parts of Rudin posted on bspace.
• 3/11: Finishing up compactness, beginning continuity, possibly review of limsup problem from midterm. Section 13, Rudin excerpt, and Section 17.
• 3/13: Continuity continued, possibly review of limsup problem from midterm. Sections 17, 18 in the book.
• 3/18: Intermediate value theorem, starting uniform continuity. Sections 18 and 19 in the book.
• 3/20: Uniform continuity, limits of functions. Sections 19 and 20 in the book.
• 3/25: Spring break!
• 3/27: Spring break!
• 4/1: Power series and uniform convergence. Sections 23 and 24 in the book.
• 4/3: More power series, Weierstrass Approximation Theorem. Sections 25 and 27 in the book (we will come back to 26).
• 4/8 MIDTERM 2 in 390 Hearst. The mean was 34.8/50. Solutions are available in Resources on bspace.
• 4/10: Weierstrass Approximation Theorem, differentiation. Sections 27 and 28 in the book, as well as a different proof of Weierstrass from Rudin on bspace.
• 4/15: Mean value theorem and other fun facts about derivatives, differentiation of power series. Sections 29 and 26 in the book; also Tim Gowers' discussion of differentiation of power series.
• 4/17: Finishing up differentiation of power series, L'Hôpital's rule, Taylor's theorem. Sections 30 and 31 in the book.
• 4/22: Finishing Taylor's theorem, introducing Riemann integration. Sections 31 and 32 in the book.
• 4/24: Darboux sums, some integration theorems, and Fundamental Theorems of Calculus. Sections 32, 33, and 34 in the book.
• 4/29: Metric spaces: continuity. Section 21 in the book.
• 5/1: Metric spaces: connectedness. Section 22 in the book.

• M. Hutchings' notes on proofs

• Homework 1, due 1/28/2014
• Homework 2, due 2/4/2014
• Homework 3, due 2/11/2014
• Homework 4, due 2/18/2014
• There is no homework due on 2/25/2014, since our first midterm is that day, but you are strongly encouraged to work through some practice problems here
• Homework 5, due 3/4/2014
• Homework 6, due 3/11/2014
• Homework 7, due 3/18/2014
• Homework 8, due 4/1/2014
• There is no homework due on 4/8/2014, since our second midterm is that day, but you are strongly encouraged to work through some practice problems here, as well as some of the metric space problems here.
• Homework 9, due 4/15/2014
• Homework 10, due 4/22/2014
• Homework 11, due 4/29/2014