Download Section 1.1: The irrationality of 2 . 1. This section introduces many of

Section 1.1: The irrationality of
2.
1. This section introduces many of the basic sets we will work with. Make sure you know exactly
what each of the following sets are: N, Z, Q, and R.
Note: Since bold letters are hard to distinguish in handwriting, we often use “blackboard bold” for
these letters: ℕ, ℤ, ℚ, and ℝ.
2. A main issue in learning to prove mathematical statements is developing a sense for when it is best
to try certain strategies. The proof that 2 is irrational is a proof by contradiction. What does this
mean? Why was this a good strategy to use?
3. The definition of a field is briefly stated on p. 3 with the comment that neither Z nor N is a field.
Why not?
4. On p. 4, Abbott comments that Q is “lacking” something. What is lacking and why is that an issue
for doing calculus?
5. The field of rational functions, R(x), can be “ordered by end behavior” in the sense that f < g if and only if
there is a real number M such that f ( x) < g ( x) for all x > M . Then R is embedded as a subfield of R(x)
as the constant functions. So Q ⊂ R ⊂ R( x) . Since the irrationals in R are described as “filling in the gaps
in Q,” how can there be even more elements in R(x)? What are these elements? What could go wrong with
doing calculus on this even larger field?
Note: Question 5 raises challenging issues that you may not be able to answer right now but should keep in
the back of your mind throughout the course.
Section 1.2: Some Preliminaries
1. This section introduces some important notation. Make sure you know how to use each of the
following: ∈ , U , I , ∅
2. Read Example 1.2.1 carefully. These three ways of defining a set will appear extensively
throughout the course, and you will need to be able to use them with ease.
3. Index notation can be confusing, but you must master it for this course! Read the examples on p. 6
carefully and make sure you know exactly what each mathematical expression means.
U
n =1
U
∞
30
Does
An include A30 ? Does
n =1
A∞ include A∞ ? Why does
U
n∈Ν
UA
∞
A∞ =
∞
?
n =1
4. Throughout the course we will use the lowercase Greek letters ε and δ to refer to real numbers
that we think of as being very small. Get good at drawing them!
Also notice that in Theorem 1.2.6, ε can represent any positive number. Explain why we only
really care about very small positive numbers in this statement. This will be common throughout
the course, so when you see a phrase like “for every positive number…” or “for every real number
ε > 0 …,” don’t be tempted to think about large numbers. To understand the material conceptually,
you want to think about very small numbers.
5. Induction can be a confusing proof strategy, but we will use it regularly throughout the course. Be
sure you understand the explanation and examples on p. 10.
Section 1.3: The Axiom of Completeness
1. The Axiom of Completeness is the conceptual and logical cornerstone of this entire course. Learn
it! Deeply. Now.
Note: We will typically refer to this as “the Least Upper Bound Property.”
2. Is the Least Upper Bound Property true for Q and R(x)? Why or why not?
3. Be prepared to flexibly use the terms “least upper bound” and “supremum” synonymously (as well
as the corresponding notation, “lub A” and “sup A”).
Section 1.4: Consequences of Completeness
1. Study the proofs of the Nested Interval Property (Theorem 1.4.1) and the Archimedean Property
(1.4.2) carefully. Not only are these important results, but you will also construct many proofs very
similar to this.
2. Does the Archimedean Property hold for R(x)? Why or why not?
3. Notice that in the proof of the Archimedean Property, we show that something exists using proof
by contradiction. Recall that in the proof that 2 is irrational, we show that something does not
exist using proof by contradiction. Explain why this is a useful technique for these two opposite
goals.
4. Sketching pictures representing your goal in a proof and key elements in steps along the way is
crucial for stringing together the logic of the full proof.
a. Sketch a number line showing the relationships between the variables used in the proof of the
Archimedean Property.
b. Study the diagram on p. 20 used to illustrate the key idea of the proof of the density of Q in R.
Make sure you understand how having just this image in mind could help you reconstruct the
entire proof.
c. Illustrate number lines for the two cases ( α 2 < 2 and α 2 > 2 ) in the proof of the existence of
square roots (Theorem 1.4.5). In each diagram, label T, α , α 2 , 2, α + 1n , and (α + 1n )2 .
5. Often when proving something, it is helpful to “work backwards” from the thing we want to prove.
The results may generate a statement to which we can more easily spot a path from our starting
point. The problem is, it then becomes crucial to keep straight which statements we have shown
are true vs. which statements we want to show are true. Read the proofs of the density of Q in R
(Theorem 1.4.3) and the existence of square roots (Theorem 1.4.5) and identify which
equalities/inequalities were generated by working backwards (thus represent a goal) and which
were deduced from the assumed information.
Chapter 1 Exam
Be able to state the following definitions: upper bound (lower bound), bounded above (bounded
below), least upper bound (greatest lower bound), least upper bound property
Be able to state and prove the following theorems: Irrationality of 2 , Nested Interval Property,
Archimedean Property, density of Q in R, existence of square roots
Be able to work and discuss proofs and examples covered in class and on homework.