Table of Contents
Lecture Notes
Lecture 3
First, we will get familiar with a few examples of convergent and non-convergent sequences. Then, we prove a few properties on convergent sequence, first, they are bounded; second, they can only converge to a single value. Then, we will prove some operations (+,-,/, etc) on convergent sequence. Finally, we prove a few useful examples of limit (Thm 9.7). See also notes from previous semester
I will present a cool trick to find limit for a recursive defined sequence, e.g. $s_{n+1}=\sqrt{s_n+1}$, for whatever initial $s_1>0$.
Discussion time: 9.2, 9.9©, 9.15
Lecture 2
Last time we recalled what is $\N$, $\Z$ and $\Q$. I hope you had realized that I did not cover much in class, and you do need to read the textbook (Ross) to learn the details. In particular, you want to work on exercises on induction, read and apply the rational zero theorem (to show that $\sqrt{2}$ is not a rational number). We also mentioned the word 'ring' and 'field', which are algebraic concepts, which you can read in Harrison's note.
Today we are going to introduce the real numbers. We are not going to construct real number from rational number. There are two ways to construct, the quickest way is 'Dedekind Cut' (you can find in Rui's note); the slower but conceptually more natural way is to 'take completion of the metric space $\Q$', which is the approach adopted by Tao-I. I am going to follow Ross, and assume the existence of real number, and state its property. One of the most important property that $\R$ have, where $\Q$ does not, is the 'existence of sup' (no, not whatsup, just sup, short for supremum). Then we prove the 'archaemedian property'. This is in Ross Sec.4
Now, you may feel all the properties in Sec 4 are too trivial. Why we spend time to prove easy properties for a familiar guy, like $\R$? Well, imagine the 'curtain of ignorance' has fallen down, and you are just given an 'ordered field with the completeness axiom', call it $F$, but you don't know yet if $F = \R$. Can you still prove the Archaemedian property? This is the axiom based approach, versus the construction based approach, it basically says, 'I don't care how you construct this field, as long as it satisfies these axioms, then it must has the following properties'.
Now, back to the proof. Can you try to prove Archaemedian property without reading the proof? Another cool statement is the denseness of $\Q$, namely, between any two real numbers, you can find a rational number. See again, if you can find a proof for that without reading the proof.
Next, we briefly mention what the symbol $+\infty$ and $-\infty$ mean. Note that, these are not real numbers, they are not member of $\R$. We introduce them to simplify certain statement of results. For example, we can now say, given any subset $E \In \R$, the $\sup(E)$ exists in $\R \cup \{+\infty\}$. (What's wrong with this expression $\R \cup +\infty$? Why the curly braces? )
That hopefully will take us 40 min. We will use the last 20 min to talk about sequence and limits in $\R$, this is Ross Sec 7 and Tao-I Ch 6. So, what does limit mean? We say a sequence (of real numbers) $(a_n)$ converges to $a$, if for any $\epsilon>0$, there exists $N>0$, such that for any $n > N$, we have $|a_n - a| < \epsilon$. Informally, we say, for any $\epsilon$, the sequence eventually fell into the $\epsilon$-neighborhood of $a$.
Let's finish by go through some examples of convergence, just to test how the definition works.
The note from previous semester might be useful.
Group Discussion (20min)
1. Let $A, B \In \R$, and define $A+B = \{a + b| a \in A, b \in B \}$. Prove that $\sup (A+B) = \sup A + \sup B$.
2. Same setup as above, prove that $\sup (A \cup B) = \max(\sup A, \sup B) $.
3. Ross 7.1, 7.2, 7.3
Lecture 1
Exercises:
- Ross 1.10, 1.12
- Read Ross 'Rational Zero Theorems', then do 2.1, 2.2, 2.7
- Try proving Ross Theorem 3.1, 3.2, by yourself, without read his proof. It is a good exercise for logical deduction. Yes, the result may sounds obvious for $\Q$, but you need to prove them for any ordered field, which you have no idea what it looks like.
Script
Welcome to Math 104, your first analysis class. You have learned about calculus, knows all about integration, perhaps also the Stokes formula, Green's formula, namely, all the useful things. What do you want to gain from this course?
In some sense, this course is a training of critical thinking. Namely, why do you believe what other people told you? Why cannot it be otherwise? Is 1+1 = 2 happens to be true in our world, or does it have to be true? (does this sentence even make sense?) Why do we want to challenge the received wisdom? (otherwise, there is no innovation). What's the benefit to challenge the received wisdom? (The invention of GPS, requires general relativity, and requires the notion of Riemannian manifold to describe curved spacetime, requires one to go beyond the Euclidean geometry, and requires one to give up the axioms that 'two parallel lines in space never intersects').
On the other-hand, it is also good to know certain dead-ends. Like, anyone knows the conservation law of energy will not attempt to invent the 'perpetual motion machine'. So, it is useful to know the rule, and where do those rules come from. And how to discover new rules. If you wish, this is like the 'classics' in literature. You are unlikely to discover something new, but the logical training you get will save you some time in the future.
Long story short, what's this class is about? As you have seen in the syllabus, there are three parts: limit, metric space topology and calculus (integration, differentiation). We will roughly spend one month each. The topic about topology might be new, and it takes some getting used to.
Today, I want to discuss about 'numbers'. By number, we have the following 3 systems $$ \N = \{0,1,2 \cdots \} $$ $$ \Z = \{ \cdots, -2,-1,0,1,2,\cdots \} $$ $$ \Q = \{ m / n \mid m, n \in \Z, n \neq 0 \} $$
$\N$ is a semi-group, with addition, but no inverse. $\Z$ is a group (abelian group), more over, it is a commutative ring, with a multiplication. $\Q$ is a 'field', with division as well.
Why do we need more? When you try to solve equation, say $x^2 = 1$ or $x^2=2$, we sometimes get two solutions, and sometimes get zero solutions in $\Q$. (the problem is that, $\Q$ is not algebraicallly closed). There is also a problem, given a set $E \In \Q$ with an upper bound, it is possible that, there is most economical 'upper bound', or sharpest upper bounded living in $\Q$. Related to this problem is that, $\Q$ is not 'complete' (namely Cauchy sequence in $\Q$ may not converge to a number in $\Q$).
Introducing the real number $\R$ solves the second problem. And there is no other option, namely, $\R$ is the unique ordered field containing $\Q$ that is complete.
Remark: there are other ways to enlarge $\Q$. What other option do you know?
Discussion questions:
1. About mathematical induction: let $P(n)$ denote a statement depending on a natural number $n$, if we can prove two things that: (a) $P(0)$ is true, and (b) $P(n)$ implies $P(n+1)$, then, we know $P(n)$ is true for all $n \in \N$.
Try Ross p6, 1.10, 1.12
2. About rational roots. Why does $x^2=2$ have no rational roots? Try Ross p12 Ex 2.2. How about 2.7?
3. There are two ways to introduce real numbers, one is through completion of $\Q$ with respect to a metric (as Tao-I) did, the other is through 'Dedekind cut' (as Ross section 6 did).
