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

• Course information

Course title: Mathematical Analysis I
Meeting times: T,Th (7–8:15pm)
Meeting place: HW-508
Office hours: T (6–7pm) in HE-924
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, ...

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 week-to-week 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
• Midterm one: Thursday, October 8th (solutions)
• Midterm two: Thursday, November 12th (solutions)
• Final exam: Thursday, December 17th, 6:20–7:40pm (solutions)

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 (2-3 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.
1. Let a/b be a rational number in reduced form. Under what conditions will √(a/b) be rational?
2. Compute the supremum and infemum of each of the following:
• {nN : n2<10}
• {n/(m+n) : m,nN}
• {n/(2n+1) : nN}
• {n/m : m,nN and m+n≤10}
3. Prove that if a is both an upper bound for A and an element of A then a is the supremum of A.
4. Use the triangle inequality to prove that if x,yRk then
||x|-|y||≤|x-y|≤|x|+|y|
5. Let de be the ordinary euclidean metric on R2, and let dm be the manhattan (diamond) metric on R2. Show that de and dm are equivalent, that is, show that A⊂R2 is open with respect to de iff it is open with respect to dm.
6. (grads) Prove the converse of 2.23, that is, if X is a metric space with a countable base then X is separable.
7. (grads) Think about problem 1.6 again if you didn't get it, and take a look at problem 1.7.
8. Prove that (A∪B)'=A'∪B'. Prove that A∪B = AB.
9. Suppose that A⊂R2 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?
10. Consider the Venn diagram on wikipedia. Prove that all the black cells are impossibilities, and give examples of each of the white cells.

2.13, (2.24)
(solutions)
• Homework 6 (Due Thursday, Oct 22)

(Some exercises from Abbott.)
1. Show that if K is a compact set of real numbers, then both sup(K) and inf(K) exist and are in K.
2. If K is compact and F is closed, prove that KF is compact.
3. Prove or give a counterexample: if F1F2F3⊃··· is a decreasing sequence of nonempty closed sets, then ∩Fn is nonempty.
4. Prove or give a counterexample: Every finite set is compact.
5. Prove or give a counterexample: Every countable set is compact.
6. Prove or give a counterexample: Every uncountable closed set of real numbers is perfect.
(solutions)
• Homework 7 (Due Thursday, Oct 29)

(Some exercises from Abbott.)
1. Prove using the definition of convergence that lim(3n+1)/(2n+5) = 3/2.
2. What happens if we reverse the order of quantifiers in the definition of convergent?

Define that xn verconges to x if there exists ε>0 such that for all NN it is true that nN implies d(xn,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?
3. Rudin, 3.4.
• Homework 8 (Due Thursday, Nov 5)

3.6abc, 3.7, 3.8, (3.19)
(solutions)
• Homework 8.5 (Fake homework)

1. 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,...}
2. 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?
3. 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?
4. Let xn≥0 for all n.
1. If xn→0 then show that (xn)1/2→0.
2. If xn≥0 and xnx then show that (xn)1/2x1/2.
5. Show that if xnynzn for all n and if xn→L and zn→L> then yn→L too.
6. What is the "length" of the set described in Abbott 3.4.4? Use the geometric series formula.
7. Rudin, 3.9.
8. Take out a calculus book and do several exercises in applying the root or ratio test. Do the same for the alternating series test.
9. 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 C1 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 a1∈C1 and b1∈C1 such that a1+b1=s.
Next, do the same for all Cn, that is, show that there exist an∈Cn and bn∈Cn such that an+bn=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
• Homework 9 (Due Tuesday, Nov 24)

1. Abbott 4.3.1. Let g(x)=∛x.
1. (warm-up) Prove that g is continuous at p=0.
2. Prove that g is continuous at p≠0. (You may need the identity a3-b3=(a-b)(a2+ab+b2).)
2. Rudin 4.2
3. Rudin 4.3
4. 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.
5. (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.)
(solutions)
• Homework 10 (Due Tuesday, Dec 8)

1. Abbott 4.3.11. Describe a function from R to R whose set of discontinuities is precisely:
1. Z
2. (0,1)
3. [0,1]
4. {1/n : n∈N}
2. Abbott 4.4.4. Show that if f is continuous on [a,b] with f(x)>0 for all axb, then 1/f is bounded on [a,b].
3. 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].
4. Rudin 4.6.
(solutions)
• Homework 10.5 (Fake homework)

1. Rudin 2.19(a),(b)
3. Rudin 3.20
4. Abbott 4.3.3. Use the ε-δ characterization of continuity to show that any linear function f(x)=ax+b is continuous at every point in R.
5. 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 AR, then any f with domain A is continuous at c.
6. Abbott 4.4.6. Give examples of each of the following, or prove that no such example exists.
1. a continuous f:(0,1)→R and a Cauchy sequence xn such that f(xn) is not a Cauchy sequence.
2. a continuous f:[0,1]→R and a Cauchy sequence xn such that f(xn) is not a Cauchy sequence.
3. a continuous f:[0,∞)→R and a Cauchy sequence xn such that f(xn) is not a Cauchy sequence.
4. a continuous and bounded f:(0,1)→R which attains a maximum value on this open interval but not a minimum value.
7. Abbott 4.5.2. Decide on the validity of the following conjectures.
1. Continuous functions take bounded open intervals to bounded open intervals.
2. Continuous functions take bounded open intervals to open sets.
3. Continuous functions take bounded closed intervals to bounded closed intervals.
8. Abbott 4.5.3. Is there a continuous function on all of R with range f(R) equal to Q?
1. Prove that for nZ, f(n)=f(1)n.
2. Prove that for qQ, f(q)=f(1)q.
3. Assume that f is continuous at 0. Prove f is continuous everywhere.
4. Using this continuity, prove that for xR, f(x)=f(1)x. )