Advanced_Calculus-Weinstein.pdf

ADVANCED CALCULUS

1996–1997

Gilbert Weinstein

Oce: CH 493B

Tel: (205) 934-3724

(205) 934-2154

FAX: (205) 934-9025

Email: weinstei@math.uab.edu

Oce hours: Monday 1:00 pm – 2:30 pm

Wednesday 8:30 am – 10:00 am

About the Course

Welcome to Advanced Calculus! In this course, you will learn Analysis, that

is the theory of dierentiation and integration of functions. However, you will

also learn something more fundamental than Analysis. You will learn what is

a mathematical proof. You may have seen some proofs earlier, but here, you will

learn how to write your own proofs. You will also learn how to understand someone

else’s proof, ﬁnd a ﬂaw in a proof, ﬁx a deﬁcient proof if possible and discard it if

not. In other words you will learn the trade of mathematical exploration.

Mathematical reasoning takes time. In Calculus, you expected to read a prob-

lem and immediately know how to proceed. Here you may expect some frustration

and you should plan to spend a lot of time thinking before you write down anything.

Analysis was not discovered overnight. It took centuries for the correct approach to

emerge. You will have to go through an accelerated process of rediscovery. Twenty

hours of work a week outside class is not an unusual average for this course.

The course is run in the following way. In these notes, you will ﬁnd Deﬁnitions,

Theorems, and Examples. I will explain the deﬁnitions. At home, on your own,

you will try to prove the theorems and the statements in the examples. You will

use no books and no help from anyone. It will be just you, the pencil and the

paper. Every statement you make must be justiﬁed. In your arguments, you may

use any result which precedes in the notes the item you are proving. You may use

these results even if they have not yet been proven. However, you may not use

results from other sources. Then, in class, I will call for volunteers to present their

solutions at the board. Every correct proof is worth one point. If more than one

person volunteer for an item, the one with the fewest points is called to the board,

ties to be broken by lot. Your grade will be determined by the number of points

you have accumulated during the term.

You have to understand the proofs presented by others. Some of the ideas

may be useful to you later. You must question your peers when you think a faulty

argument is given or something is not entirely clear to you. If you don’t, I most

probably will. When you are presenting, you must make sure your arguments are

clear to everyone in the class. If your proof is faulty, or you are unable to defend

it, the item will go to the next volunteer, you will receive no credit, and you may

not go up to the board again that day. We will work on the honor system, where

you will follow the rules of the game, and I will not check on you.

CHAPTER 1

Introduction

1. Mathematical Proof

What is a proof? To explain, let us consider an example.

Theorem1.1. There is no rational number r which is a square root of 2.

the Theorem of Pythagoras, a segment of length p 2 can be constructed as the

hypotenuse of a right triangle with both sides of length 1.

Before we prove this theorem, in fact before we prove any theorem, we must

understand its statement. To understand its statement, we must understand each

of the terms used. For instance: what is a rational number ? For this we need a

deﬁnition.

Definition1.1. A number r is rational if it can be represented as the ratio of

two integers:

(1.1)

r = n

where m 6= 0.

Of course, in this deﬁnition, we are using other terms that need to be deﬁned,

such as number, ratio, integer. We will not dwell on this point, and instead assume

for now that these have been deﬁned previously. However, already one point is clear.

If we wish to be absolutely rigorous, we must begin from some given assumptions.

We will call these axioms. They do not require proof. We will discuss this point

further later. For the time being, let us assume that we have a system of numbers

where the usual operations of arithmetic are deﬁned.

Next we need to deﬁne what we mean by a square root of 2.

Definition1.2. Let y be a number. The number x is a square root of y if

x 2 = y.

Again, we assume that the meaning of x 2 is understood. Note that we have

said a square root, and not the square root. Indeed if x 6= 0 is a square root of

y, then −x is another one. Note that then, one of the two numbers x and −x is

positive. Now, we may give the proof of Theorem 1.1.

Proof. The proof is by contradiction. Suppose that x is a square root of 2,

and that x is rational. Clearly, x 6= 0, hence we may assume that x > 0. Then,

x 2 = 2, and there are integers n, m 6= 0, such that

x = n

m .

This theorem was already known to the ancient Greeks. It was very important

to them since they were particularly interested in geometry, and, as follows from

(1.2)

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