M361K Assignment 1
due 6/2/06
Welcome to M361K, Introduction to Real Analysis. In this course, we will be rigorously
establishing properties of the real line, functions, and notions of calculus. By “rigorously
establishing properties”, what we mean is that we will be providing statements (theorems)
and arguments that rigorously and logically establish their validity (proofs). This is the fun-
damental process of mathematics. It may take us a while to get used to proving theorems.
We will be practicing our ability to develop, evaluate and understand proofs throughout
this class, both in our writing and our speaking.
In the homework, you will see several problems to work on. You are responsible for
writing up solutions to each problem. Those problems with an asterix (*) next to them
denote problems that will be presented by you, the students, to the class. More on that in
a bit. In the meantime, let’s talk a bit about this whole theorem/proof business.
A theorem is a statement along with a proof. A theorem is not considered valid unless
there is a proof that we, the mathematicians, agree is convincing. Even if there is over-
whelming evidence that a theorem is true (e.g. it has been confirmed in a thousand cases),
it is not considered true unless a proof is given. Note that there is no absolute test for
truth of a theorem, only a consensus by the mathematical community of its validity. So
when we are explaining a proof, the validity of the proof is determined by the audience, in
which case we need to make sure that our level of explanation is sufficient for the audience.
A professor explaining a proof to another professor may sound different than a professor
explaining a proof to a group of first year math students. For this class, when presenting
proofs we shall assume that our audience is a capable group of undergraduate math stu-
dents familiar with basic notions of mathematics, but this is the first time they have heard
of the theorem you are presenting. Of course a purported theorem (a conjecture) can be
shown to be false with the presentation of a single example. Let’s see what this all might
look like in practice.
Let’s suppose that your friend Sarah says that she has discovered the following two part
theorem.
Sarah’s Theorem.
A) The sum of any even integer and any odd integer is an odd integer.
B) The product of any even integer and any odd integer is an odd integer.
Sarah, like ALL OF US, makes mistakes so you decide the best way to check this out is
to try to prove the theorem yourself. For A) you first try to understand the statement by
considering some examples: 5 + 4 = 9, 11 + 20 = 31, 2 + 1 = 3. To prove the statement
you need to know exactly what it says. This means knowing the definitions. What is an
even integer? You could say “an even integer is any of the numbers 2,4,6,8,....” Is this
correct? What about 0? Is 0 even? What about −2 or −10? Are they even? Consulting
the definition provided you see that they are. Reading the statement of A) carefully we
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