## Basic Real Analysis. Math 3310, Section 1. Spring 2015

### TuTh 2-3:15 pm, Monroe 111.

• Instructor: Mikhail Ershov
• Office: Kerchoff 302
• e-mail: ershov at virginia dot edu
• Office hours: Tue 3:30-4:30, Wed 2:15-3:15 (Clark 101), Wed 4:45-5:45 by APPOINTMENT
• Course webpage:   http://people.virginia.edu/~mve2x/3310_Spring2015

### Course outline

In this course we will rigorously introduce the main concepts of real analysis, including limits, continuity, differentiability, integrability and series, and prove the main theorems dealing with them (e.g. Intermediate Value Theorem, Mean Value Theorem and the Fundamental Theorem of Calculus). While all these topics and theorems are discussed in detail in Calculus I and II, not much is usually being said about their proofs. The goal of this course is not only (and not so much) to convince you that these theorems are indeed true, but rather to introduce you to the basic techniques used to prove those theorems and teach you how to apply those techniques to solve other problems. In particular, you will learn how to work with the ``epsilon-delta'' definition of the limit which is a fundamental skill for understanding any advanced topics in analysis. The plan is to cover the majority of material from Chapters 1-6 of Wade's book. I will assume basic familiarity with sets and functions (Section 1.5). You should read this section on your own in the first couple of weeks and ask me questions about unclear points.

A note on proofs. MATH 3310 is a proof-based course. Everything we discuss in class will be rigorously proved. More importantly, you are expected not only to understand proofs, but also to learn how to construct your own proofs and how to write proofs  (so that others can understand your argument). It is not expected that you have taken a proof-based course before MATH 3310. However, MATH 3310 will not include any lectures devoted specifically to proof writing; instead you will be expected to develop this skill gradually as we progress through the material. If you feel that you need a more detailed introduction to proof writing, I encourage you to take MATH 3000 (Transition to Higher Mathematics) before or concurrently with MATH 3310. MATH 3000 is offered this semester and will meet MWF 10-10:50am. An assessment exam will be administered at the beginning of the semester to help you make the decision about MATH 3000 .

### Preliminary schedule.

 week sections topics Jan 13, 15 1.1, 1.2 Introduction. Ordered field axioms. Jan 20, 22 1.3 Least upper bounds. Completeness axioms. Jan 27, 29 2.1, 2.2 Limits of sequences. Limit theorems. Feb 3, 5 2.3, 2.4 Bolzano-Weierstrass Theorem. Cauchy sequences Feb 10, 12 1.6, 3.1 Countable and uncountable sets. Limits of functions. Feb 17, 19 3.2, First midterm. One-sided limits. Limits at infinity. Feb 24, 26 3.3 Continuity. Mar 3, 5 3.4, 4.1 Uniform continuity. The derivative. Mar 17, 19 4.2, 4.3 Differentiability Theorems. Mean Value Theorem. Mar 24, 26 4.4 Taylor's Theorem and L'Hospital's Rule. Mar 31, Apr 2 4.5, Second midterm. Inverse function theorems. Apr 7, 9 5.1, 5.2 The Riemann integral. Riemann sums. Apr 14, 16 5.2, 5.3 More on Riemann sums. The Fundamental Theorem of Calculus. Apr 21, 23 6.1, 6.2 Introduction to series. Series with non-negative terms. Apr 30 6.3 Absolute convergence.

### Evaluation

The course grade will be based on homework, two midterms and the final (all in-class), with weights distributed as follows:
• homework: 15%
• midterms: 20% each
• final: 45%

### Exams

The midterm dates given below are tentative and may be changed later. The date and time of the final exam is determined by the registrar and cannot be changed.
• First midterm exam: Thu, Feb 19th
• Second midterm exam: Thu, Apr 2nd
• Final exam: Mon, May 4th, 9am-12pm

### Make-up policy

• Make-ups will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up before the exam.
• If you miss an in-class exam or fail to submit a take-home exam by the due date without a compelling reason, you will be assigned the score of 0 on that exam.
• University regulations specifically prohibit early make-ups.

### Homework

• Homework will be assigned weekly and will usually be due on Thursday.
• No late homework will be accepted. However, three lowest homework scores will be dropped.
• GRADING: it will not be possible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit.

### Collaboration policy on homework.

• You are welcome (and even encouraged) to work on homework together, but you must write up solutions independently, in your own words. In particular, you should not be consulting others during the process of writing down your solution .

### Announcements

Major announcements will be made in class and also posted on the course webpage. Some other announcements may only be made by e-mail, so check your e-mail account regularly.

• Monday, January 26 -- Last day to ADD a course, elect the audit option, change the grading option (grade or CR/NC), or establish an independent study
• Tuesday, January 27 -- Last day to DROP a course (deletion from the transcript)
• Wednesday, March 18 -- Last day to withdraw from a course