# Math 104

Section 2 - Fall 2006### Overview

**Lectures:** Tu, Th 9:30 - 11:00am, Room **70** Evans

**Instructor:** Jan Reimann

**Office:** 705 Evans Hall

**Office hours:** Tu 11 - 1, We 1-2, and by appointment

**Discussion session:** We 4:30-6pm

**Email:**

**Personal Website:** http://math.berkeley.edu/~reimann/

### Course Description

In this course we will study the foundations of real analysis. This means we will get acquainted with the real number system, how it can be defined axiomatically. We will then use the basic properties of the real numbers to study fundamental notions of analysis such as *sequences* and *series*. A thorough understanding of these is a prerequisite for any higher level mathematics.

We will also see how this notions can be treated in a more general spaces, not necessarily based on real numbers, namely *metric spaces*. This will give us a first glimpse at how mathematics uses generalization.

Next, we study functions over the real numbers. The basic concept here is *continuity*. In particular, we will learn how to describe the intuitive concept of continuity ("a function without jumps") formally using the notion of sequences, and prove results about continuity rigorously. As continuity is a purely *topological* concept, we will see that we can treat it much more generally in metric spaces.

After that, we will start approximating functions by other functions. This brings us to *sequences and series* of functions. A fundamental way to approximate is using *power series*, which can be thought of as "infinite" (or better, sequences of) polynomials.

In the final part of the course we will revisit concepts already known from Calculus, *differentiation* and *integration*. This time, we will investigate them from another angle, much less interested in doing actual calculations than in proving basic properties.

Throughout the class, strong emphasis will be put on **writing rigorous mathematical proofs**.

### Literature

The textbook for the course will be *K.A. Ross, Elementary Analysis: The Theory of Calculus*. We will follow it more or less closely. Chapters 1-4 will be the core topics of the course, and so we will study them most thoroughly. The last part of the course will cover parts of chapters 5 and 6.

I will occasionally draw material from two other texts, Rudin's *Principles of Mathematical Anaylsis* and *Real Mathematical Analysis* by Pugh, especially supplements on metric spaces and examples.

### Exams

There will be **two midterms**: the first on **Tuesday, September 26** (in class), the second on **Tuesday, November 7** (also in class). Note that the first midterm is before the five-week drop-out deadline.

**Information on the first midterm (09/26):** This will be a **closed book exam**(sorry, **no cheat sheets!**). Bring your **blue books**. There is a sample midterm available.

**Information on the second midterm (11/07):** Again, this will be a **closed book exam**(sorry, **no cheat sheets!**). The material covered will be **Ross, §§ 13-15, 17-21, 23-26**. Bring your **blue books**. There is a sample second midterm available.

The **final exam** is held on **Thursday December 14, 8-11 am, 166 Barrows**. As before, a **closed book exam** (**no cheat sheets**). The final exam will cover all material covered in class (see log below), with an emphasis on the second part of the semester. Bring your **blue books**. There is a sample final available.

### Grading

The final grade will be determined as follows: Only yhe better one of the two midterm scores will count towards the final grade. The final grade will be the better one calculated by the following two methods: ** (1) 20% homework, 40% midterm score, 40% final exam**; **(2) 1/3 homework, 1/3 midterm, 1/3 final**.

### Homework

Homework will be assigned on Tuesday and will be **due on the
following Tuesday** in class. Homework will be graded and the
lowest score will be dropped. Late homework will not be accepted.

A note on **academic honesty**: Collaboration among
students to solve homework assignments is welcome. This is a good way
to learn mathematics. So is the consultation of other sources such as
other textbooks. However, *every student should hand in an own set
of solutions*, and if you use other people's work or ideas you
*should indicate the source in your solutions*.

(In any
case, complete and correct homework receives full credit.)

Date | File | Due | Solutions |
---|---|---|---|

09/05 | Homework 1 (pdf) | 09/12 | |

09/12 | Homework 2 (pdf) | 09/19 | |

09/19 | Homework 3 (pdf) | 09/26 | |

10/03 | Homework 4 (pdf) | 10/10 | |

10/10 | Homework 5 (pdf) | 10/17 | |

10/17 | Homework 6 (pdf) | 10/24 | |

10/24 | Homework 7 (pdf) | 10/31 | |

10/31 | Homework 8 (pdf) | 11/07 | |

11/14 | Homework 9 (pdf) | 11/21 | |

11/29 | Homework 10 (pdf) | 12/05 |

### Graduate Student Instructor

There is a GSI for this course, **Aubrey Clayton**. His office hours are: *Mo 10-12, 2-5* and
*Tu 9:30-11, 1:30-5* in *891 Evans*.

### Brief Summary of Lectures

Date | Material Covered | Suggested Textbook Reading |
---|---|---|

08/29 | Introduction; the rational numbers; Pythagoras' proof of the irrationality of square roots; approximations by rational numbers; the field axioms |
Chap. 1, § 1-3 |

08/31 | Consequences of the field axioms; discussion of the notion of an axiom; orders and ordered fields; properties of ordered fields |
Chap. 1, § 3 |

09/05 | lower and upper bounds, infima and suprema; completeness; the real numbers as a complete ordered field; consequences of completeness: the Archimedean property, density of the rationals; absolute value and distance function |
Chap. 1, § 3,4, and 6 |

09/07 | absolute value and distance function; sequences of real numbers; convergence; examples of convergent and divergent sequences |
Chap. 1, § 3, 6; Chap. 2, § 7, 8 |

09/12 | natural numbers and induction; stability properties of limits; examples; monotone and bounded sequences |
Chap. 1, § 1; Chap. 2, § 8, 9, 10 |

09/14 | lim sup and lim inf; Cauchy sequences; subsequences |
Chap. 2, § 10, 11 |

09/19 | Bolzano-Weierstrass Theorem; cardinality; finite, countable and uncountable sets |
Chap. 2, § 11; Rudin Chap. 2, pp. 24 - 30 |

09/21 | comparing cardinalities; Cantor-Schroeder-Bernstein Theorem; Cantor's diagonal method and uncountability of the set of reals |
Rudin Chap. 2, pp. 24 - 30 |

09/26 | 1st Midterm | |

09/28 | Review midterm; Infinite series; convergence; Cauchy criterion for series; geometric and harmonic series |
Chap. 2, § 14 |

10/03 | Tests for convergence of series: comparison, root, ratio test; alternating series; integral tests; metric spaces | Chap. 2, § 13, 14, 15 |

10/05 | Examples of metric spaces; convergence and completeness in metric spaces; topology; open balls, open sets, closed sets, examples | Chap. 2, § 13 |

10/10 | Characterizations of open and closed sets; interior, boundary, and limit points; the Cantor set; compactness; examples; statement of the Heine-Borel Theorem |
Chap. 2, § 13 |

10/12 | Proof of the Heine-Borel Theorem |
Chap. 2, § 13 |

10/17 | Continuous functions, stability properties; Continuous functions on compact sets; Intermediate value theorem |
Chap. 3, § 17, 18 |

10/19 | Applications of the Intermediate Value Theorem; uniform continuity; uniform continuity and compactness; continuity in arbitrary metric spaces |
Chap. 3, § 18-21 |

10/24 | Power series, radius of convergence; sequences of functions; pointwise and uniform convergence; uniform convergence and continuity |
Chap. 4, § 23-25 |

10/26 | uniform convergence of power series; Abel's Theorem' Weierstrass approximation theorem; | Chap. 4, § 26-27 |

10/31 | exponential function; complex numbers; definition and properties of sin and cos; | |

11/02 | properties of trigonomeric functions; review of material for midterm; | |

11/07 | second midterm | |

11/09 | review of second midterm; differentiability; | Chap. 5, § 28 |

11/14 | chain rule for differentiation; mean value theorem | Chap. 5, §§ 28,29 |

11/16 | mean value theorem; examples | Chap. 5, § 29 |

11/21 | Taylor series | Chap. 5, § 31 |

11/28 | Definition of the Riemann integral | Chap. 6, § 32 |

11/30 | Properties of the Riemann integral | Chap. 6, § 33 |

12/05 | Fundamental theorem of calculus | Chap. 6, § 33 |

12/07 | Integration and differentiation of power series | Chap. 4, § 26 |