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Real Analysis
Workbook
Compiled by: Jerry Morris, Sonoma State University
Note to Students (Please Read): This workbook contains examples and exercises that
will be referred to regularly during class. Please print out the rest of the workbook before our next class and bring it to class with you every day.
- To Print Out the Workbook. Go to the Moodle page for our course and click on the link “Work- book”, which will open the file containing the workbook as a .pdf file. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINT- ERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK.
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Table of Contents
Introductory Material
Sample Proofs.................................................................................................. 3 Proof Writing Standards....................................................................................... 5
Chapter 0 –
Sections 0.1& 0.2 – Sets, Relations, and Functions.............................................................. 8 Section 0.3 – Mathematical Induction......................................................................... 16 Section 0.4 – Equivalent and Countable Sets.................................................................. 18 Section 0.5 – The Real Numbers.............................................................................. 24
Chapter 1 –
Section 1.1 – Sequence Convergence........................................................................... 28 Section 1.2 – Cauchy Sequences............................................................................... 35 Section 1.3 – Arithmetic Operations on Sequences............................................................. 42 Section 1.4 – Subsequences and Monotone Sequences.......................................................... 46
Chapter 2 –
Section 2.1 – Limits........................................................................................... 53 Section 2.2 – Limits of Functions and Sequences............................................................... 60 Section 2.3 – The Algebra of Limits........................................................................... 65
Chapter 3 –
Section 3.1 – Continuity of a Function at a Point.............................................................. 70 Section 3.2 – Algebra of Continuous Functions................................................................ 74 Section 3.3 – Uniform Continuity and Compactness........................................................... 77 Section 3.4 – Properties of Continuous Functions.............................................................. 87
Chapter 4 –
Section 4.1 – The Derivative of a Function.................................................................... 92 Section 4.2 – The Algebra of Derivatives...................................................................... 97 Section 4.3 – Rolle’s Theorem and the Mean Value Theorem................................................. 101
Chapter 5 –
Section 5.1 – The Riemann Integral.......................................................................... 108 Section 5.2 – Classes of Integrable Functions................................................................. 116 Section 5.3 & 5.4 – Riemann Sums and the Fundamental Theorem of Calculus............................... 120
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Why Do We Prove?
After reading the proofs on the previous page, you might be wondering why we bother to prove such “obvious” statements as those on the previous page. You might also wonder over the course of the semester why I am so picky about the details in your proofs, especially those that seem geometrically obvious to you. One reason is that, in mathematics, there are lots of statements that sound plausible, even obvious, and yet turn out to be false; the only way to convince the world that such statements are false is by constructing a careful, rigorous proof of this fact. As an example, consider the following statements, EACH OF WHICH CAN BE PROVEN FALSE when made mathematically precise in a natural way.
Statement Comment If you take a convergent infinite series and rear- range the order of the terms, the sum of the series must stay the same.
False! If a series is conditionally convergent, one can change the value of the sum by rearranging the order of terms. One can even make the series diverge by changing the order of the terms.
The set of rational numbers is bigger than the set of integers.
False! The set of integers and the set of rational numbers are exactly the same size. We’ll prove that later this semester. If you start with the interval [0, 1] and remove in- tervals whose lengths add up to 1, there will be no points left.
False! In fact, it is possible to remove intervals from [0, 1] whose lengths add up to 1 in such a way that there are infinitely (even uncountably!) many points remaining. Every function f : R −→ R that is continuous everywhere must be differentiable almost every- where.
False! There exist functions that are continuous everywhere but nowhere differentiable. You can’t graph them, and it’s difficult to picture them, but you can prove that they exist!
Real Analysis Workbook 5
Comments on Proofs
Introduction.
Just what is a proof, anyway? This is a hard question to answer, in general, since it depends on your level of experience and on the expectations of those reading your proofs. In this course, our proofs will start out simple, even “obvious” in some cases, with the goal being to write a convincing proof without logical gaps. Later on, the results we prove will be more complicated and you will have progressed in your sophistication to the point where you can say “clearly” more often without losing credit for not showing all of the details. Let’s start, then, with the all-important question:
What can I assume without proof, and what do I need to actually show? While this is a
difficult question to answer in general, here are some guidelines:
- You may assume basic arithmetic facts (involving addition, subtraction, multiplication, and division) when doing calculations in proofs that involve real numbers. In particular, any of the 12 axioms in Section 0.5, as well as Theorems 0.19, 0.20, and 0.21 may be used without citation at any time in your proofs. For example, let’s say that you know that 3 − 2 n > 7. It would be perfectly valid for the following sequence of arguments to appear in your proof: 3 − 2 n > 7 =⇒ − 2 n > 4 =⇒ n < − 2. Though various axioms of the real numbers and basic theorems are being used to get from one step to another, no citations of the specific axioms or theorems are necessary. Just show enough intermediate steps so that your logic is clear.
- You may use any theorems from our text, results of homework problems from our text, or results given in class as justification for conclusions made in your proofs, as long as they appear sequentially before the particular problem you are doing, and as long as the directions of the problem don’t rule out the use of a particular result you are using. If you do make use of various theorems, exercises, or facts from class, they should be appropriately cited, unless they’re excluded by point number 1 above.
- In general, results of homework problems and theorems from other texts MAY NOT be used in your proofs. If you find yourself in a situation that you feel might warrant an exception to this general rule, feel free to discuss it with me.
- When in doubt about whether or not to include a detail, it’s best to include it!
- Finally, remember that if you’re ever in doubt as to whether or not something can be assumed or needs to be shown, please feel fee to ask me.
Other Points. Here are a few general things I’ll be looking for in your proofs:
- Your proofs should contain mostly words and should read in complete sentences. Using accepted mathematical notation and symbols is okay, but it shouldn’t be so excessive that it interrupts the flow of the proof.
- When you cite a theorem as part of your proof, it is your obligation to verify that the hypotheses of the theorem are satisfied as part of your proof.
- Keep in mind that logical correctness is not the only issue in judging how good a proof is; the quality of the communication is also crucial. For example, if a proof is complicated and convoluted to the point that it takes your reader a calendar year to figure out that it is valid, then it’s not a very good proof!
- In order for your proof to be clear, it is important to put the supporting facts for a claim at the appropriate places in the proof so it is clear to the reader which fact justifies which claim. Hopefully the following discussion will clarify what I mean:
Real Analysis Workbook 7
The last point above is a complicated one, since it is sometimes difficult to tell when you are jumping to conclusions. To help you out, here are a couple of examples. Let’s start with the following definitions and then try to prove the theorem that follows them.
Definition 1. We say that an integer n is even if n = 2k for some integer k.
Definition 2. We say that an integer n is odd if n = 2k + 1 for some integer k.
Theorem. The sum of any two odd integers is even.
Here are some examples of sample proofs of this statement. Some are better than others. Point out any weaknesses you notice in these proofs.
Proof 1. Consider the odd integers 3 and 7. Then 3 + 7 = 10, which is even. Since this logic works for any odd integers, the sum of any two odd integers is always even.
Proof 2. Let m and n be any odd integers. Since m and n are both odd, m = p + 1, where p is even, and n = q + 1, where q is even. Therefore, m + n = p + q + 1 + 1 = p + q + 2. Since p and q are both even, p + q + 2 must also be even.
Proof 3. Let m and n be any odd integers. Since m and n are both odd, there exists integers p and q such that m = 2p + 1 and n = 2q + 1. Thus, we have m + n = (2p + 1) + (2q + 1) = 2p + 2q + 2, which is clearly divisible by 2. Thus, m + n is even.
Proof 4. Let m and n be any odd integers. Since m and n are both odd, there exists integers p and q such that m = 2p + 1 and n = 2q + 1. Thus, we have m + n = (2p + 1) + (2q + 1) = 2(p + q + 1). Since p + q + 1 is an integer, it follows that m + n is even.
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Sections 0.1 & 0.2 – Sets, Relations, and Functions
Some Special Sets:
• J = { 1 , 2 , 3 , 4 ,.. .}
• Z = {... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,.. .}
- ∅ = the empty set
- R = {real numbers}
- [a, b] =
- (a, b) =
Some Definitions and Notation: Let A and B be sets.
- We say that A is a subset of B (written A ⊂ B) if every element in A is also in B.
- The union (A ∪ B) and intersection (A ∩ B) of the sets A and B are defined below:
A ∪ B = {x : x ∈ A or x ∈ B} A ∩ B = {x : x ∈ A and x ∈ B}
- The complement of A relative to B is the set
B\A = {x : x ∈ B and x 6 ∈ A}
Examples.
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Example. Calculate each of the following.
- (a) {[0, λ]}λ∈{ 1 , 2 , 3 }
(b)
λ∈{ 1 , 2 , 3 }
[0, λ]
(c)
λ∈{ 1 , 2 , 3 }
[0, λ]
- (a) {(−n, n)}n∈J
(b)
n∈J
(−n, n)
(c)
n∈J
(−n, n)
Real Analysis Workbook 11
- (a)
λ∈{ 2 , 4 , 6 }
{λ − 1 , λ + 1}
(b)
λ∈{ 2 , 4 , 6 }
{λ − 1 , λ + 1}
⋂^ ∞
n=
n
n
x∈Q
{x}
- For each n ∈ J, let An =
−∞, − (^1) n
]
[ 1
n ,^ ∞
(a)
⋃^ ∞
n=
An
(b)
⋂^ ∞
n=
An
Real Analysis Workbook 13
Definition. Let f : A −→ B.
- We say that f is one-to-one or injective if f (x) = f (y) always implies that x = y.
- We say that f maps A onto B if im(f ) = B; i.e., if for every b ∈ B, there exists a ∈ A such that f (a) = b.
Example. Which of the following functions are one-to-one? onto?
(a) f : R −→ R defined by f (x) = 3x − 1
(b) f : R −→ R defined by f (x) = x^2
(c) f : R −→ R defined by f (x) = sin x
(d) f : [− π 2 , π 2 ] −→ R defined by f (x) = sin x
(e) f : [− π 2 , π 2 ] −→ [− 1 , 1] defined by f (x) = sin x
Example. For each of the following functions, illustrate f and its converse, g.
(a) f : R −→ R defined by f (x) = 2x + 5
(b) f : Z −→ Z defined by f (x) = x^2
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Theorem 1. Let f be a function, and define
g = {(x, y) : (y, x) ∈ f }.
Then g is a function iff f is one-to-one. If g is a function, g is called the inverse of f and we write g(x) = f −^1 (x).
Theorem 2. Let f : A −→ B and g : B −→ C be functions.
- If f and g are both one-to-one, then g ◦ f is one-to-one.
- If f maps A onto B, and g maps B onto C, then g ◦ f maps A onto C.
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Section 0.3 – Mathematical Induction
First Principle of Mathematical Induction. Let P (n) represent a proposition depending on n. If
- P (1) is true, and
- P (k) implies P (k + 1) for each integer k ≥ 1 ,
then P (n) is true for all n ∈ J.
Second Principle of Mathematical Induction (Strong Induction). Let P (n) represent a proposition depending on n, and let m be a positive integer. If
- P (1), P (2),... , P (m) are all true, and
- for all integers k ≥ m, the truth of the statements P (1), P (2),... , P (k) implies the truth of P (k + 1),
then P (n) is true for all n ∈ J.
Real Analysis Workbook 17
Examples and Exercises
- Prove by induction that 1^3 + 2^3 + 3^3 + · · · + n^3 = n^2 (n + 1)^2 4
- Define f : J −→ Z by f (1) = 3, f (2) = 5, and f (n) = 3f (n − 1) − 2 f (n − 2) for n ≥ 3. Prove that f (n) = 1 + 2n^ for all n ∈ J.
Real Analysis Workbook 19
Definition 1. Let A and B be any sets. We say that A is equivalent to B if there exists a function
f : A −→ B such that f is both one-to-one and onto. If A is equivalent to B, we write A ∼ B.
Example 1. Let D = { 1 , 3 , 5 , 7 ,.. .}. Prove that J ∼ D.
Definition 2.
- S is finite means either S = ∅ or S ∼ { 1 , 2 , 3 ,... , n}.
- S is infinite means S is not finite.
- S is countably infinite means.
- S is countable means that either S is or.
- S is uncountable means that S is not countable.
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Example 2. Is Z countable? Justify your answer.
Facts About Countable Sets
Fact 1. Subsets of countable sets are countable.
Fact 2. Let f : S −→ J be one-to-one. Then S is countable.
