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This workbook contains examples and exercises that will be referred to regularly during class. It covers topics such as sets, relations, functions, mathematical induction, real numbers, sequence convergence, limits, continuity, derivatives, and integrals. The workbook also includes sample proofs and proof writing standards. It is compiled by Jerry Morris from Sonoma State University. Students are advised to print out the workbook before class and bring it every day.
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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.
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Sample Proofs.................................................................................................. 3 Proof Writing Standards....................................................................................... 5
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
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
Section 2.1 – Limits........................................................................................... 53 Section 2.2 – Limits of Functions and Sequences............................................................... 60 Section 2.3 – The Algebra of Limits........................................................................... 65
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
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
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
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:
difficult question to answer in general, here are some guidelines:
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
A ∪ B = {x : x ∈ A or x ∈ B} A ∩ B = {x : x ∈ A and x ∈ B}
B\A = {x : x ∈ B and x 6 ∈ A}
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(b)
λ∈{ 1 , 2 , 3 }
[0, λ]
(c)
λ∈{ 1 , 2 , 3 }
[0, λ]
(b)
n∈J
(−n, n)
(c)
n∈J
(−n, n)
Real Analysis Workbook 11
λ∈{ 2 , 4 , 6 }
{λ − 1 , λ + 1}
(b)
λ∈{ 2 , 4 , 6 }
{λ − 1 , λ + 1}
n=
n
n
x∈Q
{x}
−∞, − (^1) n
n ,^ ∞
(a)
n=
An
(b)
n=
An
Real Analysis Workbook 13
(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
(a) f : R −→ R defined by f (x) = 2x + 5
(b) f : Z −→ Z defined by f (x) = x^2
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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).
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Section 0.3 – Mathematical Induction
First Principle of Mathematical Induction. Let P (n) represent a proposition depending on n. If
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
then P (n) is true for all n ∈ J.
Real Analysis Workbook 17
Examples and Exercises
Real Analysis Workbook 19
f : A −→ B such that f is both one-to-one and onto. If A is equivalent to B, we write A ∼ B.
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Facts About Countable Sets