### MATH 401/501 - HOMEWORK PROBLEMS - Spring 2014

Numbered exercises are from Tao's Book. For these problems you can only assume results and exercises prior to the given exercise (if you are going to use a previous result, make sure you state it, of course if it has not been proved in class and is not one of the other homework problems, you can attempt to prove those intermediate results, but this is not required.)

Homework 11 (last) (due on Thursday May 1st, if you turn it in on Tuesday April 29 you get 5 bonus points).

1. Section 10.1: Exercises 10.1.4(g), 10.1.5.
2. Section 10.2: Exercise 10.2.2 (Practice 10.2.3)
3. Section 10.3: Exercise 10.3.3 (Practice 10.3.2)
4. Section 10.4: Exercise 10.4.1.

BONUS HOMEWORK (due anytime before the end of Review week on May 8th)

1. Section 4: Exercise 9.4.7
2. Section 5: Exercise 9.5.1
3. Section 7: Exercise 9.7.2
4. Section 8: Exercise 9.8.4.

Homework 10 (due on Thursday April 24 (if turned in on Tuesday April 22, 5 bonus points)

1. Section 9.3: Exercise 9.3.1 (Practice 9.3.2, 9.3.5)
2. Section 9.4: Exercise 9.4.2 (examples) (Practice 9.4.1, 9.4.3, 9.4.4)
3. Section 9.6: Exercise 9.6.1 (examples)
4. Section 9.8: Exercise 9.8.2 (examples) (Practice 9.8.1) (BONUS: 9.8.5)
5. Section 9.9: Exercise 9.9.6 (composition preserves uniform continuity)

Homework 9 (due Thursday April 17 - you get 5 bonus points if turned in on Tuesday April 15):

1. Section 7.5: Exercises 7.5.1 (an inequality comparing liminf of ratio and nth root), 7.5.2 (a specific power series) and 7.5.3 (these are examples).
2. Section 9.1: Exercise 9.1.3 (for this you need Definition 9.1.10 of the Closure of a Set: the collection of all adherent points to E; and you need Definition 9.1.8 of adherent points to a set: x in R is an adherent point to a set E in R iff for all epsilon >0 there is y in E such that |y-x|< epsilon.
3. Section 9.2: Exercise 9.2.1 (about real valued functions defined on R).
4. Bonus: Exercise 7.4.1
5. Reading Assignment: Sections 7.4-7.5 and Sections 9.1-9.2

Homework 8 (due Thursday April 3 - you get 5 bonus points if turned in on Tuesday April 1):

1. Section 6.3: Exercise 6.3.1 (Practice 6.3.4).
2. Section 6.4: Exercises 6.4.5 (squeeze test), 6.4.6 and 6.4.9 (build examples) (Practice Exercise 6.4.4, 6.4.9, BONUS 6.4.10.
3. Section 6.5: Exercise 6.5.2 (Practice 6.5.1).
4. Section 6.6: Exercises 6.6.2 (build an example), 6.6.4 (subsequences vs limits).
5. Section 6.7: BONUS Exercise 6.7.1

Homework 7 (due Thursday 3/27/14, 5 bonus points if you turn this in on Tuesday 3/25/14):

1. Section 5.5: Exercise 5.5.1
2. Section 5.6: Exercise 5.6.2(a)(c) (Practice 5.6.1, BONUS: 5.6.2(b)(d)(e))
3. Section 6.1: Exercise 6.1.5 (convergent in R implies Cauchy in R),
Exercise 6.1.8 (b) (e) (limit laws) (BONUS: 6.1.6, 6.1.8(g)) [Practice: 6.1.4, 6.1.10]

Homework 6 (due Thursday 3/13/14, 5 bonus points if you turn this in on Tuesday 3/11/14):

1. Section 5.3: Exercises 5.3.2 (Multiplication is well defined, I did half of this exercise in class), 5.3.5 (LIM{1/n}=0) (Practice 5.3.3)
2. Section 5.4: Exercise 5.4.4 (an Archimedean Principle) (BONUS: 5.4.3 and 5.4.7) (Practice: 5.4.2, 5.4.5)

Homework 5 (due Thursday 3/6/14, 5 bonus points if you turn this in on Tuesday 3/4/14):

1. Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded)
2. Section 5.2: Exercise 5.2.1 (equivalent sequences if one Cauchy so is the other)

Homework 4 (due Thursday 2/20/14, 5 bonus points if you turn this in on Tuesday 2/18/14):

1. Section 4.1: Exercises 4.1.2 and 4.1.7(d) (Practice 4.1.3, 4.1.4, 4.1.8)
2. Section 4.2: Exercise 4.2.6 (Practice 4.2.2, 4.2.7)
3. Section 4.3: Exercise 4.3.2(f)(g)
4. Section 4.4: BONUS Exercise 4.4.1

Homework 3 (due Th Feb 13, 2014, if turned on Tue Feb 11 get 5 bonus points):

1. Section 3.1: Exercise 3.1.10 (Practice: 3.1.6 and 3.1.8))
2. Section 3.3: Exercise 3.3.7 (Practice 3.3.2 and 3.3.5))
3. Section 3.4: Exercise 3.4.2 (Practice 3.4.1 and 3.4.5))
4. Section 3.5: Exercise 3.5.3 (Practice 3.5.4))
5. Section 3.6: Exercise 3.6.5 (BONUS: 3.6.9 or 3.6.6 to be turned in separately to me)

Homework 2 (due Th Feb 6, 2014, if turned on Tue Feb 4 get 5 bonus points):

1. Exercise 2.3.1 (p.32 2nd ed, p.36 1st ed) (multiplication is commutative)
2. Exercise 2.2.3 (d) OR (f) [p.29 2nd ed, p.33 1st ed] (Properties of order)
3. Exercise 2.3.5 [p.32 2nd ed, p.36 1st ed] (Euclidean Algorithm). Also show that the m,r you find are unique.

Homework 1 (due on Th Jan 30, 2014, if turned in on Tue Jan 28 get 5 bonus points):
For the first two problems assume arithmetic and whatever you know about inequalities (turn on your number knowledge):

1. Show by induction that for all n natural numbers 1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.
2. Show by induction that n^2 <= 2^n for all natural numbers n>3.
3. Show that n+0 = 0+n (here you turn off your knowledge, all you know is the Peano axioms and the definition of addition).
Hint: prove it by induction.

Quiz 1 solved and more.