Sets, Numbers, and Sequences: An Introduction to Mathematical Proofs, Study Guides, Projects, Research of Mathematics

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Introduction to Proof in Analysis - 2020 Edition

Contents

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 *Equivalence Relations

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 1

Introduction

1.1 Purpose

Mathematics explores a universe inspired by, but different from, the real world we live in. It is full of wonderfully beautiful phenomena, but whose truth can only be validated by rigorous logical arguments, which we call proofs. The purpose of this course is to introduce you to this universe, to help you learn and apply the language and techniques of mathematical proof, and in the process to prepare you for Math 410.

Becoming familiar with a new language can be a frustrating process, espe- cially when it is not simply a matter of translation of English into a different language. There are two fundamental differences:

First, Mathematical language is how we describe and validate mathematical phenomena (different from what we use normal language for) and so we have to learn to understand the phenomena we are describing.

Second, in normal spoken language we often speak ”approximately” and our audience can easily figure out what we mean from the context. In Mathematics it is essental to use the language absolutely precisely, because it is the unique tool we have to validate the truth of our assertions.

As you come to grips with this course I would strongly encourage you to take full advantage of my availability to support your efforts outside this class.

Here are some of the ways available to you. You can do this individually or in groups!

• Ask questions in class!!
• Ask questions before or after class!!
• Come to office hours!!
• Make appointments to see me outside office hours!!
• Email me with questions!!

Now this course may seem quite different from your previous math courses, where a major focus may have been on how to construct derivatives and inte- grals, to multiply and invert matrices, and to solve linear equations.

I would like to illustrate that difference with an example of a proof of a theo- retical mathematical result, which still only uses facts you may have seen in your calculus classes. For this example we recall that [c,d] denotes the closed interval of real numbers x satisfying c ≤ x ≤ d. Similarly, [c,a), (a,b) and b,d] denote respectively the intervals of real numbers x satisfying c ≤ x < a, a < x < b and b < x ≤ d.

Example: Recall that a real polynomial of degree n is a real-valued function of the form f (x) = a 0 + a 1 x + · · · + anxn,

in which the ak are real constants and an 6 = 0. A real zero of such a polynomial is a real number b such that f (b) = 0.

Theorem: Suppose such a real polynomial f (x) of degree n and with an = 1 has n distinct real zeros, b 1 < ... < bn.

Let [c, d] be an interval such that c < b 1 < ... < bn < d,

Then there is a positive number ε > 0 such that if g(x) is any differentiable function for which |g(x)| < ε, for every x ∈ [c, d],

then f (x) + g(x) has at least n zeros in [c, d].

Proof: We do this in Steps

Step One: f (x ) = (x − b 1 )(x − b 2 )...(x − bn ).

You may already have seen this, but in any case I am going to prove it a little later, and so will skip the proof for now.

Step Two: In each of the intervals [c, b 1 ), (b 1 , b 2 ), ..., (bn , d ], f (x )is never zero and therefore has constant sign.

Then for each i, f (yi)+g(yi) has the same sign as f (yi). Therefore, f (yi)+g(yi) and f (yi+1) + g(yi+1) have opposite signs.

Therefore, by the Intermediate Value Theorem, there are points zi in the intervals (yi, yi+1) such that

(f + g)(zi) = 0.

q.e.d.

1.2 Expectations:

What you can expect from me:

1. A clear statement of what material will be covered, of what you will be expected to learn to do, and of how that will be assessed:

(a) The material to be covered is explicitly stated in the Syllabus. (b) You will be expected to learn to solve problems and express the so- lutions as formal, clearly written mathematical proofs.

You will not be asked to repeat proofs of theorems and definitions.

However, unless you know these cold you will not be able to pro- duce correctly written solutions. (c) Assessment will be through weekly homework assignments, 3 term tests, and a final exam. Your work will be graded on how well you meet the following requirements:

Your answers must be a sequence of statements, each following log- ically from the previous ones, with an explanation as to why. They must be

• clear, precise, and unambiguous,
• written in complete, grammatical sentences, and
• for homeworks only, typed or written legibly in ink.

They may rely only on:

• any statement already proved in class;
• any statement proved in an earlier part of the text;
• any statement in any earlier exercise.

2. Good access to me outside lectures

(a) Office hours are posted in the Syllabus.

(b) I am often in my office outside office hours, and you are always wel- come to drop in and see if I am free to answer a question or explain something. (c) You can always make an appointment by email to see me. (d) I regularly check my email and respond to questions and requests (but not too late at night!).

3. Readiness to respond to questions, and to clarify material you have not understood.

(a) It is almost a mathematical theorem that working with me outside class can be a big factor in your success. (b) My basic philosophy is: There are no stupid questions! (c) I will always try to help you when you are stuck, but I can be more useful to you if you have first tried to unstick yourself.

What you should expect from yourselves:

1. Stay engaged with the course.
2. Be proactive: always ask a question if you do not understand something, or do not see how to attack a problem.
3. Join a study group if that is helpful, but never as a freeloader.
4. Be prepared to work at finding a solution to a problem: problems are not just variations on examples from class.
5. Master and be comfortable with the meaning of the words and sym- bols as they are introduced.
6. Stay current with the material, and do the assigned readings as assigned.
7. Do the assigned homework each week.

Students who invest this time and effort usually do well. Students who do not make this effort usually do not.

2. Let....: This is a statement which defines some terminology or estab- lishes notation. As an example: Let ”‘n”’ be a natural number greater than 5.
3. Theorem, proposition, lemma: These are names for mathematical statements that are going to be proved.

To summarize: it is essential that you internalize the definitions and no- tation rather than simply memorize them. Indeed, when learning to speak or write a new language you need to be able to use the words spontaneously with- out having to call up each corresponding English word and then translate it. In the same way, you need to be able to speak/write mathematics, not just remember the dictionary of definitions. This is essential, because

When you write a statement in Mathematics it must say to any reader exactly what you mean

Finally, Mathematics uses a number of logical expressions for you to become familiar with, since they are how the deductions in a proof are expressed. Ev- eryone, even at an early age, learns what it means for a statement to follow logically from others, and sometimes it will be obvious that the statement is correct. So, writing proofs and solving problems does not require you to learn to think logically, since you already know how to do that. What it does require is for you to learn how to use that ability in the world of Mathematics.

2.2 What is a Proof in Mathematics?

A proof in Mathematics is a sequence of statements which establish that certain assumptions (the hypotheses) imply that a certain statement (the conclu- sion) is true. This sequence of statements must satisfy:

1. Each is clear and unambiguous,
2. Each is true, and its truth follows immediately from the truth of the preceding statements and the hypotheses.
3. The final statement is the conclusion.

Example 1 1. Prove that if a student is in Math 310 (the hypothesis), s/he is registered at the University of Maryland(the conclusion).

Proof: To be in Math 310 you must be registered at the University. Therefore, since the student is in Math 310, s/he is registered at the Uni- versity. q.e.d.

2. Prove that if x is a real number and x > 1 then x^2 > x.

Proof: x^2 − x = x(x − 1). Since x > 1 , x and x − 1 are both positive. Therefore x(x − 1) is positive. Therefore x^2 > x. q.e.d.

3. If n is a natural number and n > 1 , prove that n^3 − 1 is divisible by n − 1.

Proof: let n be a natural number and suppose n > 1. Direct multipli- cation gives (n − 1)(n^2 + n + 1) = n^3 − 1. Since n^2 + n + 1 is a natural number, n^3 − 1 is divisible by n − 1. q.e.d.

2.3 To solve a 310 problem

Here are the basic steps

1. Fully understand the problem, and the mathematical terms used. In particular, make sure you understand

(a) what you are given (the hypotheses); (b) what you are asked to prove (the conclusion); (c) the terminology and the notation.

2. Figure out why the statement to prove is true.
3. Express your solution in a formal mathematical proof, as follows:

(a) Start your solution with ”We have to prove that ...” This is what is to be proved. (b) Then, beginning with the word ”proof ” provide a sequence of sen- tences, each following logically from the preceding ones and the hy- potheses. (c) Write so that each sentence has a single meaning which will be clear to any reader. (d) The final statement should be the conclusion, which your proof has now established as true. (e) Note: When writing mathematics it is almost always useful to la- bel the objects under consideration. This is called ”establishing notation”.

Tips for Writing Proofs from Elizabeth (a former Math 310 student):

1. Learning to write proofs is not like learning how to multiply matrices, and it can take mental effort and time.

Definition 2 Numbers

1. N will denote the set of natural numbers, and we write N = { 1 , 2 , 3 ,.. .}. Thus 0 is not a natural number.
2. Z will denote the set of integers: Z = { 0 , ± 1 , ± 2 , ± 3.. .}.
3. Q will denote the set of rational numbers: these are the real numbers which can be written p/q with p ∈ Z and q ∈ N. Note that p/q = r/s if and only if ps = rq.
4. R will denote the set of real numbers.
5. C will denote the set of complex numbers.
6. An integer n is even if it is divisible by 2 ; ii is odd if it is not even.
7. A prime number is a natural number n which is larger than 1 and is not divisible by any natural number except 1 and n.
8. If n ∈ N then n is positive and −n is negative. If a = p/q ∈ Q and p ∈ N then a is positive and −a is negative.
9. A real number x is larger than a real number y if x − y is positive. (CAUTION: Note that − 1 is larger than − 2 !)
10. If x is a real number and n is a natural number then xn^ is the real number obtained by multiplying x with itself n times.
11. The absolute value of a real number x is denoted by |x| and is defined by: |x| =

x, x ≥ 0 , −x, x < 0 Note that |x| ≥ 0 for all x ∈ R.

12. A real number, x, is positive if x > 0 and negative if x < 0. Thus 0 is neither positive or negative.

For Exercise 1 below you may find it helpful to read Sec. 2.6 in the text.

Exercise 1 1. Show that for each rational number a ∈ Q exactly one of the following three possibilities is true:

(a) a > 0 (b) a = 0 (c) a < 0

2. If a, b, c are any three rational numbers, show that

(a) If a < b and b < c then a < c.

(b) a + b < a + c if and only if b < c. (c) Multiplication by a positive preserves inequalities. (d) Multiplication by a negative reverses inequalities.

3. If a ∈ Q then a > 0 if and only if a > 1 /n for some n ∈ N.

Proposition 1 The Division Proposition for Numbers: If p < n are natural numbers then for some natural number m ≤ n and for some integer r with 0 ≤ r < p, n=mp +r.

(The number r is called the remainder.)

Proof: Let p < n be natural numbers. If k > n then for any r ≥ 0

kp ≥ k > n.

Thus if mp + r = n it must be true that mp ≤ n and so m ≤ n. Since 1 ≤ n we can choose m to be the largest natural number such that mp ≤ n. Then n − mp ≥ 0.

On the other hand, if n − mp ≥ p then (n − mp) − p > 0 and so

n − (m + 1)p = (n − mp) − p > 0.

Thus m was not the largest natural number which multiplied by p gave an answer at most n, and we assumed it was. Thus n − mp < p and we may set r = n − mp. q.e.d.

Definition 3 A sequence starting at k ∈ Z is a list (xn)n≥k of objects, pos- sibly with repetitions, and indexed by the integers n, n ≥ k. If (xn)n≥k is a sequence, then xn is the nth^ term in the sequence.

Note: The objects listed in a sequence form a set, as seen in the first two of the following examples.

Example 3

1. The list 0 , 1 , 0 , , 1 , 0 , 1 , · · · is a sequence. The corresponding set is the collection { 0 , 1 } which only has two elements.
2. The list 2 , 4 , 2 , 4 , 6 , 2 , 4 , 6 , 8 , 2 , 4 , 6 , 8 , 10 , · · · is a sequence. The correspond- ing set is the collection of even natural numbers.
3. A sequence (xn)n≥ 1 of numbers is defined by

xn = 2n^ + n^2.

(a) Every student has a height h such that h ≤ a where a is the height of the tallest student. (b) Every student has a height h such that every student is in a class.

2.5 Sums, Products, and the Sigma and Pi No-

tation

1. The ”Sigma” notation is used for sums: If vi are numbers or vectors (or anything else we know how to add) then ∑^ n

i=

vi = v 1 + v 2 + · · · vn.

More generally, ∑^ n

i=

vki = vk 1 + vk 2 + vk 3 + · · · vkn.

In this notation the ”i” just indexes the terms being added or multiplied, and we could use any letter instead without changing the meaning. Thus

∑^ n

i=

vi =

∑^ n

l=

vl =

∑^ n

q=

vq = v 1 + v 2 + · · · vn.

In other words, the ”i” is a dummy variable just as in calculus, where we have (^) ∫ f (x)dx =

f (u)du

2. The ”Pi” notation is used for products: If vi are numbers then ∏^ n

i=

vi = v 1 · v 2 · · · vn.

More generally, ∏n

i=

vki = vk 1 · vk 2 · vk 3 · · · vkn.

3. In particular, if C is a constant, then ∑^ n

i=

C = C + · · · + C (n times) = nC

and (^) n ∏

i=

C = C · · · C = Cn.

Example 4 Prove that if every student in Math 310 gets at least 7 / 10 on a homework then the class average is at least 7 / 10.

Proof: Let the number of students in the class be n (Establishes notation.). Suppose every student has a grade at least 7 / 10. Let xi be the grade of the ith student. Then, by definition, the class average is ∑n i=1 xi n

Since each xi ≥ 7 , therefore

∑n i=1 xi n

∑n i=1 xi n

7 n n

∑n i=1(xi^ −^ 7) n

Therefore the class average is at least 7 / 10. q.e.d.

Theorem 1 (Difference Theorem) For any non-zero real numbers a, b ∈ R and any n ∈ N,

bn^ − an^ = (b − a)

n∑− 1

i=

bian−^1 −i.

Proof: Note that

b(

n∑− 1

i=

bian−^1 −i) =

n∑− 1

i=

bi+1an−^1 −i^ =

n∑− 1

j=

bj+1an−^1 −j^.

In the second expression set j + 1 = i.Then

n∑− 1

j=

bj+1an−^1 −j^ =

∑^ n

i=

bian−i,

and so

b(

n∑− 1

i=

bian−^1 −i) =

∑^ n

i=

bian−i^ = bn^ +

n∑− 1

i=

bian−i.

On the other hand,

(−a)(

n∑− 1

i=

bian−^1 −i) = −

n∑− 1

i=

bian−i^ = −an^ −

n∑− 1

i=

bian−i.

Adding these two lines gives the equation of the Theorem. q.e.d.

3. However, the statement ”for every natural number there is a natural num- ber that is smaller” is false, since there is no natrural number smaller thatn 1.
4. let S be a set, and let

(a) A be the statement ” Every x ∈ S has property P, and let (b) B be the statement ” Some x ∈ S has Property P.

Then A implies B but B does not imply A.

Exercise 3

1. If A ⇒ B and B ⇒ C does A ⇒ C?
2. Construct three statements, A,B,C in which A implies B, A implies C, but it is not true that B implies C. Prove that each of your assertions is correct.
3. Is the following statement true or false? Provide a proof for your answer. ”Suppose A and B are natural numbers, and that A = 5 if and only if B = 2. If B 6 = 2, then A = 7.
4. What is the contrapositive of the following statements?

(a) A 6 = 8 if B = 9. (b) A = 8 if B 6 = 9.

5. What is the converse of the following statements? (a) B = 9 if A = 8. (b) A 6 = 8 if B = 9.
6. Show that the statemen ”A ⇒ B” is true if and only if its contrapositive is true.

2.7 Examples of Mathematical Statements and

their Proofs

Example 6

1. Lemma Let (xn) be the sequence of rational numbers defined by xn = 1 − 1 /n. Then for each k ∈ N there is some natural number N such that | 1 − xn| < 1 / 2 k if n ≥ N.

Proof: Fix a natural number k. Then choose N = 3k. Thus if n ≥ N , then 1 /n ≤ 1 /N , and | 1 − xn| = | 1 − (1 − 1 /n)| = 1/n ≤ 1 /N = 1/ 3 k < 1 / 2 k.

q.e.d.

2. This example illustrates the importance of using words carefully and correctly.

Let (xn) be the sequence of integers defined by xn = (−1)n. Consider the following two assertions

(a) Statement: There is an ε ∈ R, and there is an N ∈ N such that |xn − 0 | < ε for n ≥ N.

(b) Statement: For each ε > 0 there is an N ∈ N such that |xn − 0 | < ε for n ≥ N.

The first is correct and the second is false.

To prove the first statement we will pick some ε ∈ R and some N ∈ N so that the conclusion is satisfied. So we pick

ε = 3 and N = 1.

Then for n ≥ N ,

|xn − 0 | = |(−1)n^ − 0 | = 1 < 3 = ε.

Now for the second statement. It states that for each ε ”something” is true. Thus to prove that it is false we have to find some ε for which that something is not true.

Here we pick ε = 1/ 2. Then for any N ∈ N, and n ≥ N ,

|xn − 0 | = 1

which is not less than 1 / 2.

3. Statement: For each real number a and for each ε > 0 there is a δ > 0 such that if |x − a| < δ then |x^2 − a^2 | < ε.

In this example δ will depend on both a and ε: a larger a and a smaller ε will require a smaller δ.

4. Statement:If x = 2 and y = 4 then x + y = 6. This assertion is true.

However, the converse assertion: ”if x + y = 6 then x = 2 and y = 4” is false, because if x = 1 and y = 5 then x + y = 6. (This is called a counterexample)

5. Lemma: Suppose x and y are natural numbers. Then xy = 2 only if either x ≥ 2 or y ≥ 2.