Samuel Coskey / Courses / Analysis 351/641, Fall 2009

Course information
Course title: Mathematical Analysis I
Meeting times: T,Th (7–8:15pm)
Meeting place: HW508
Office hours: T (6–7pm) in HE924
Textbook: Walter Rudin,Principles of Mathematical Analysis
, 3rd ed 
Collaboration
I would like for everybody to join the group http://groups.google.com/group/math351hunter! We will use this site for announcements and discussions about course topics and problems. 
Topics
Chapters 1–4 of Rudin. The real number system, set theory, real topology, metric spaces, convergence, sequences and series of real numbers, functions, continuity, ... 
Grading
I will give three exams—two midterms and a final. Each exam will be worth about one quarter of your grade. The last quarter of the grade will come from homework, and other projects. I will assign and collect homework on a weektoweek basis, usually on Thursday. That makes about eleven homeworks, out of which I'll drop the lowest. 
Exam dates
I never could get the hang of Thursdays.
—Arthur Dent 
Graduate version
Some homework problems and some test questions (in parentheses) will be required for the graduate students, but extra/optional for the undergraduates. Hence, the graduate students will be assigned final grades independently from the undergraduates. 
Essay
Please write a short essay (23 pages) and turn it in by Tuesday, November 3. Choose one of the topics suggested below. You may also create your own, but please check with me to make sure it's OK. Read the following two articles on math education. Discuss the positions of the two authors. How are they related? Are their ideas compatible?
 Read the following article in favor of math history. Then pick your favorite theorem or area of pure mathematics, research its history, and tell me the whole story. Finally, give some ways in which knowing the story changed your view of the theorem or area.
 Think of an industry which uses mathematics. Tell me exactly what aspects of pure mathematics are used in this industry, and how they are used. The following article gives some examples. Finally, describe how the mathematics that you discussed fits into the college curriculum. Does college prepare one to go into this industry? Does it need to? Why?

Homework 1 (Due Thursday, Sept 10)
1.1, 1.2, 1.4, 1.5, (1.6), 1.8
(solutions) 
Homework 2 (Due Thursday, Sept 17)
1.9, 1.11, 1.12, 1.13, (1.17), 1.18
(solutions) 
Homework 3 (Due Thursday, Sept 24)
2.2, 2.3, 2.4, 2.5, (2.6), 2.11
(solutions) 
Homework 4 (Due Thursday, Oct 1)
2.6, 2.8, 2.9aef, 2.10, (2.22), (2.23)
(solutions) 
Homework 4.5 (Fake homework)
Don't turn this in, just use it to study for the midterm. I reserve the right to expand or edit the list over the next couple of days. Let a/b be a rational number in reduced form. Under what conditions will √(a/b) be rational?

Compute the supremum and infemum of each of the following:
 {n∈N : n^{2}<10}
 {n/(m+n) : m,n∈N}
 {n/(2n+1) : n∈N}
 {n/m : m,n∈N and m+n≤10}
 Prove that if a is both an upper bound for A and an element of A then a is the supremum of A.

Use the triangle inequality to prove that if
x,y∈R^{k} then
xy≤xy≤x+y  Let d_{e} be the ordinary euclidean metric on R^{2}, and let d_{m} be the manhattan (diamond) metric on R^{2}. Show that d_{e} and d_{m} are equivalent, that is, show that A⊂R^{2} is open with respect to d_{e} iff it is open with respect to d_{m}.
 (grads) Prove the converse of 2.23, that is, if X is a metric space with a countable base then X is separable.
 (grads) Think about problem 1.6 again if you didn't get it, and take a look at problem 1.7.
 Prove that (A∪B)'=A'∪B'. Prove that A∪B = A∪B.
 Suppose that A⊂R^{2} has isolated points. Can it be open? What if A⊂X where X is the space from problem 2.10, and A has isolated points. Can A be open?
 Consider the Venn diagram on wikipedia. Prove that all the black cells are impossibilities, and give examples of each of the white cells.

Homework 5 (Due Thursday, Oct 15)
2.13, (2.24)
(solutions) 
Homework 6 (Due Thursday, Oct 22)
(Some exercises from Abbott.) Show that if K is a compact set of real numbers, then both sup(K) and inf(K) exist and are in K.
 If K is compact and F is closed, prove that K∩F is compact.
 Prove or give a counterexample: if F_{1}⊃F_{2}⊃F_{3}⊃··· is a decreasing sequence of nonempty closed sets, then ∩F_{n} is nonempty.
 Prove or give a counterexample: Every finite set is compact.
 Prove or give a counterexample: Every countable set is compact.
 Prove or give a counterexample: Every uncountable closed set of real numbers is perfect.
 (Grads: 2.16)

Homework 7 (Due Thursday, Oct 29)
(Some exercises from Abbott.) Prove using the definition of convergence that lim(3n+1)/(2n+5) = 3/2.

What happens if we reverse the order of quantifiers in the
definition of convergent?
Define that x_{n} verconges to x if there exists ε>0 such that for all N∈N it is true that n≥N implies d(x_{n},x)<ε.
Give an example of a vercongent sequence. Can you give an example of a vercongent sequence that is divergent? What exactly is being described in this strange definition?  Rudin, 3.4.
 (Grads: Rudin, 3.3)

Homework 8 (Due Thursday, Nov 5)
3.6abc, 3.7, 3.8, (3.19)
(solutions) 
Homework 8.5 (Fake homework)

Abbott 3.3.5. Which of the following sets are compact? Why or why not?
 Q
 Q∩[0,1]
 R
 Z∩[0,10]
 {1,1/2,1/3,1/4,1/5,...}
 {1,1/2,2/3,3/4,4/5,...}
 Abbott 3.4.4(a). Repeat the Cantor construction starting with the interval [0,1] and at each step removing the open middle fourth from each component. Is the resulting set compact? Perfect?

Abbott 2.2.4. Argue that the sequence
1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,(5 zeroes),1,...
does not converge to 0. For what values ε>0 does there exist a response N? For which values of ε>0 is there no suitable response? 
Let x_{n}≥0 for all n.
 If x_{n}→0 then show that (x_{n})^{1/2}→0.
 If x_{n}≥0 and x_{n}→x then show that (x_{n})^{1/2}→x^{1/2}.
 Show that if x_{n}≤ y_{n}≤ z_{n} for all n and if x_{n}→L and z_{n}→L> then y_{n}→L too.
 What is the "length" of the set described in Abbott 3.4.4? Use the geometric series formula.
 Rudin, 3.9.
 Take out a calculus book and do several exercises in applying the root or ratio test. Do the same for the alternating series test.

Grads: Abbott 3.3.6. Let C be the cantor set and show that
C+C={a+b  a,b∈C} is equal to the entire interval
[0,2]. To do this, let C_{1} be the nth set in the
construction of the Cantor set, shown in Abbott on page 76.
Show that given any s∈[0,2], there exist
a_{1}∈C_{1} and
b_{1}∈C_{1} such that
a_{1}+b_{1}=s.
Next, do the same for all C_{n}, that is, show that there exist a_{n}∈C_{n} and b_{n}∈C_{n} such that a_{n}+b_{n}=s.
Finally, take a subsequential limit to reach the desired conclusion, namely that there exist a and b in the Cantor set such that a+b=s  (Grads: Rudin, 3.23.)

Abbott 3.3.5. Which of the following sets are compact? Why or why not?

Homework 9 (Due Tuesday, Nov 24)
 Abbott 4.3.1. Let g(x)=∛x.
 (warmup) Prove that g is continuous at p=0.
 Prove that g is continuous at p≠0. (You may need the identity a^{3}b^{3}=(ab)(a^{2}+ab+b^{2}).)
 Rudin 4.2
 Rudin 4.3
 Abbott 4.3.8(a). Show that if a function is continous on R and equal to 0 at every rational point, then it must be identically 0 on all of R.
 (Grads: Abbott 4.3.12. Let C be the Cantor set. Let g be the function from [0,1] into R defined by g(x)=1 if x∈C, and g(x)=0 if x∉C. Prove that g is discontinuous at every point p∈C, and g is continuous at every point p∈[0,1]\C.)
 Abbott 4.3.1. Let g(x)=∛x.

Homework 10 (Due Tuesday, Dec 8)

Abbott 4.3.11. Describe a function from R to R
whose set of discontinuities is precisely:
 Z
 (0,1)
 [0,1]
 {1/n : n∈N}
 Abbott 4.4.4. Show that if f is continuous on [a,b] with f(x)>0 for all a≤x≤b, then 1/f is bounded on [a,b].
 Abbott 4.5.7. Let f be a continuous function on the closed interval [0,1] with range also contained in [0,1]. Prove that f must have a fixed point; that is, show f(x)=x for at least one value of x∈[0,1].
 Rudin 4.6.
 (Grads: Rudin 4.19)

Abbott 4.3.11. Describe a function from R to R
whose set of discontinuities is precisely:

Homework 10.5 (Fake homework)
 Rudin 2.19(a),(b)
 Rudin 2.20, see also Abbott 3.4.7
 Rudin 3.20
 Abbott 4.3.3. Use the εδ characterization of continuity to show that any linear function f(x)=ax+b is continuous at every point in R.

Abbott 4.3.4. Show that any function f with domain Z is
necessarily continuous at every point in Z.
Show in general that if c is an isolated point of A⊂R, then any f with domain A is continuous at c. 
Abbott 4.4.6. Give examples of each of the following, or prove
that no such example exists.
 a continuous f:(0,1)→R and a Cauchy sequence x_{n} such that f(x_{n}) is not a Cauchy sequence.
 a continuous f:[0,1]→R and a Cauchy sequence x_{n} such that f(x_{n}) is not a Cauchy sequence.
 a continuous f:[0,∞)→R and a Cauchy sequence x_{n} such that f(x_{n}) is not a Cauchy sequence.
 a continuous and bounded f:(0,1)→R which attains a maximum value on this open interval but not a minimum value.

Abbott 4.5.2. Decide on the validity of the following
conjectures.
 Continuous functions take bounded open intervals to bounded open intervals.
 Continuous functions take bounded open intervals to open sets.
 Continuous functions take bounded closed intervals to bounded closed intervals.
 Abbott 4.5.3. Is there a continuous function on all of R with range f(R) equal to Q?
 (Grads: Rudin 4.5)

(Grads: Abbott 4.3.10. Let f:R→R satisfy
the additive condition
f(x+y)=f(x)+f(y).
 Prove that for n∈Z, f(n)=f(1)n.
 Prove that for q∈Q, f(q)=f(1)q.
 Assume that f is continuous at 0. Prove f is continuous everywhere.
 Using this continuity, prove that for x∈R, f(x)=f(1)x. )