**Math 310-0101**: Introduction to Proof in Analysis - Spring 2020** Revised Mar 25**

**Instructor: **Professor Steve Halperin

**Contact Information**:

Email: shalper@umd.edu.

Office: Only available by Zoom.

Phone: 301-405-1875 (from any campus
phone dial 58175) Note: This is unavailable while I am not allowed on campus

Course webpage: http://www.math.umd.edu/~

Key to Success: A fundamental strategy in this course is to ask questions: in class, during office hours, or by email. The resulting student-teacher exchange can be critical to the learning process.

**Textbook**:
The textbook is posted at Text.

You may download it for your **personal use**
**only.**

**Outline of the Course:**

The course will cover most of the material in the online textbook as described below:

Chapter 1 Introduction

Purpose

Expectations

Chapter 2 Mathematical Proofs

The Language of Mathematics

What is a Proof in Mathematics?

Solving a 310 Problem

Sets, Numbers, and Sequences

Sums, Products, and the Sigma and Pi Notation

Logical Expressions for Proofs

Examples of Mathematical Statements and their Proofs

The True or False Principle: Negations, Contradictions, and

Counterexamples

Proof and Construction by Induction

Polynomials

The Literature of Mathematics

Chapter 3 Basic Set Theory

Sets

Operations with Sets

Maps between Sets

Composites, the Identity Map, and Associativity

Onto, 1-1, and 1-1 Correspondences

Chapter 4 The Real Numbers

Properties of the Rational Numbers

The Real Numbers, Inequalities, and the Sandwich Theorem

Absolute Value

Bounds

Least Upper and Greatest Lower Bounds

Powers

Constructing the Real Numbers

Chapter 5 Infinite Sequences

Convergent Sequences

Bounded Sequences

The Cauchy Criterion for Convergence

The Intersection Theorem

Subsequences

Chapter 6 Continuous Functions of a Real Variable

Real-valued Functions of a Real Variable

Limits

Limits and Negations

Limits of Sequences and Limits of Functions

Continuous Functions

Continuous Functions Preserve Intervals

Additionally, during the week I am usually available via Zoom and students are welcome make a Zoom appointment using the procedure you will receive shortly!

**Lecture Classroom** on line via Zoom.. The Zoom meetings are already scheduled.

Calculators:

**Classroom rules:** These will be sent soon.

**Homework:**
Homework must be typed or written in ink. All
homework problems will be taken from the exercises in the
text and will be assigned in the Assignments section of Zoom and/or on the web at Homework, Midterms, and Exam with a specified due date. It must be uploaded via Zoom bu midnight.

Help Sessions: TBD

Students missing a test or exam will receive a grade of zero unless they have requested and received from me in writing an approved absence. For students with an approved absence the term test component of the final grade will be computed from the other two test grades.

Approvals will normally be granted only in the following circumstances: religious observances; mandatory military obligations; serious family or medical issues; or conflicts with other university requirements.

**Midterm Schedule: **Midterm 1: Fri. Feb. 21 in class** **

**
Midterm 2: Fri. Apr. 10 NOTE Change !!**

Time and Date (Tentative) Mon May 18, 8 AM-10 AM

**Cheating:** Students who cheat will be prosecuted according to the
university regulations

Campus undergraduate student/course policies and procedures: http://www.ugst.umd.edu/

**Instructor: **Professor Steve Halperin

**Contact Information**:

Email: shalper@umd.edu.

Office: second floor of the Math Building, Room
2107.

Phone: 301-405-1875 (from any campus
phone dial 58175) Note: This is unavailable while I am not allowed on campus

Course webpage: http://www.math.umd.edu/~

Key to Success: A fundamental strategy in this course is to ask questions: in class, during office hours, or by email. The resulting student-teacher exchange can be critical to the learning process.

**Textbook**:
The textbook is posted at Text.

You may download it for your **personal use**
**only.**

**Outline of the Course:**

The course will cover most of the material in the online textbook as described below:

Chapter 1 Introduction

Purpose

Expectations

Chapter 2 Mathematical Proofs

The Language of Mathematics

What is a Proof in Mathematics?

Solving a 310 Problem

Sets, Numbers, and Sequences

Sums, Products, and the Sigma and Pi Notation

Logical Expressions for Proofs

Examples of Mathematical Statements and their Proofs

The True or False Principle: Negations, Contradictions, and

Counterexamples

Proof and Construction by Induction

Polynomials

The Literature of Mathematics

Chapter 3 Basic Set Theory

Sets

Operations with Sets

Maps between Sets

Composites, the Identity Map, and Associativity

Onto, 1-1, and 1-1 Correspondences

Chapter 4 The Real Numbers

Properties of the Rational Numbers

The Real Numbers, Inequalities, and the Sandwich Theorem

Absolute Value

Bounds

Least Upper and Greatest Lower Bounds

Powers

Constructing the Real Numbers

Chapter 5 Infinite Sequences

Convergent Sequences

Bounded Sequences

The Cauchy Criterion for Convergence

The Intersection Theorem

Subsequences

Chapter 6 Continuous Functions of a Real Variable

Real-valued Functions of a Real Variable

Limits

Limits and Negations

Limits of Sequences and Limits of Functions

Continuous Functions

Continuous Functions Preserve Intervals

Additionally, during the week I am usually available via Zoom and students are welcome make a Zoom appointment using the procedure you will receive shortly!

**Lecture Classroom** on line via Zoom.. The Zoom meetings are already scheduled.

Calculators:

**Classroom rules:** These will be sent soon.

**Homework:**
Homework must be typed or written in ink. All
homework problems will be taken from the exercises in the
text and will be assigned in the Assignments section of Zoom and/or on the web at Homework, Midterms, and Exam with a specified due date. It must be uploaded via Zoom bu midnight.

Help Sessions: TBD

Students missing a test or exam will receive a grade of zero unless they have requested and received from me in writing an approved absence. For students with an approved absence the term test component of the final grade will be computed from the other two test grades.

Approvals will normally be granted only in the following circumstances: religious observances; mandatory military obligations; serious family or medical issues; or conflicts with other university requirements.

**Midterm Schedule: **Midterm 1: Fri. Feb. 21 in class** **

**
Midterm 2: Fri. Apr. 10 NOTE Change !!**

Time and Date (Tentative) Mon May 18, 8 AM-10 AM

**Cheating:** Students who cheat will be prosecuted according to the
university regulations

Campus undergraduate student/course policies and procedures: http://www.ugst.umd.edu/