Course information

Collaboration

Topics

Grading

Exam dates

• Midterm one: Thursday, October 8th (solutions)
• Midterm two: Thursday, November 12th (solutions)
• Final exam: Thursday, December 17th

Graduate version

Essay

• Read the following two articles on math education.

  A mathematician's lament
  Why are we learning this?

  Discuss the positions of the two authors. How are they related? Are their ideas compatible?
• Read the following article in favor of math history.

  Why 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.

  Examining how mathematics is used in the workplace

  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)

Homework 2 (Due Thursday, Sept 17)

Homework 3 (Due Thursday, Sept 24)

Homework 4 (Due Thursday, Oct 1)

Homework 4.5 (Fake homework)

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:
   {n∈N : n²<10}
   {n/(m+n) : m,n∈N}
   {n/(2n+1) : n∈N}
   {n/m : m,n∈N 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,y∈R^k then
   ||x|-|y||≤|x-y|≤|x|+|y|
5. Let d_e be the ordinary euclidean metric on R², and let d_m be the manhattan (diamond) metric on R². Show that d_e and d_m are equivalent, that is, show that A⊂R² is open with respect to d_e iff it is open with respect to d_m.
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 = A∪B.
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.

Homework 5 (Due Thursday, Oct 15)

Homework 6 (Due Thursday, Oct 22)

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 K∩F is compact.
3. Prove or give a counterexample: if F₁⊃F₂⊃F₃⊃··· is a decreasing sequence of nonempty closed sets, then ∩F_n 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.
7. (Grads: 2.16)

Homework 7 (Due Thursday, Oct 29)

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 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?
3. Rudin, 3.4.
4. (Grads: Rudin, 3.3)

Homework 8 (Due Thursday, Nov 5)

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&cap[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 x_n≥0 for all n.
   (a) If x_n→0 then show that (x_n)^1/2→0.
   (b) If x_n≥0 and x_n→x then show that (x_n)^1/2→x^1/2.
5. oops, duplicate of previous.
6. 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.
7. What is the "length" of the set described in Abbott 3.4.4? Use the geometric series formula.
8. Rudin, 3.9
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.
10. 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 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
11. Grads: Rudin, 3.23.

Homework 9 (Due Tuesday, Nov 24)

1. Abbott 4.3.1. Let g(x)=∛x.
   (a) (warm-up) Prove that g is continuous at p=0.
   (b). 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.