### MATH 401/501 - HOMEWORK PROBLEMS - Fall 2017

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.)

Homwork 11(due on Tuesday December 5th). Do three of the following problems, the rest will cound as bonus points if you choose to try them.

1. Section 10.2: Exercise 10.2.5 (Mean Value Theorem from Rolle's Theorem).
2. Section 10.2: Exercises 10.2.6 and 10.2.7 (functions with bounded derivative are Lipschitz and are uniformly continuous).
3. Section 11.1: Exercise 11.1.4 (on common refinement of partitions).
4. Let f,g: [a,b]-->R be bounded functions. Show that sup{f(x)+g(x):x in [a,b]} <= sup{f(x): x in [a,b]} + sup{g(x): x in [a,b]}. What happens when you replace sup by inf?
5. Section 11.4: Exercise 11.4.1(a)(b)(d)(e) (Linear laws and monotonicity for Riemann integration. Use the Riemann sum definitions for upper and lower Riemann integrals in Def 11.3.9 and Prop 11.3.12. Exercises 3 and 4 will help for part (a)).
6. Section 11.9: Exercise 11.9.2 (antiderivative for the same function differ by a constant).
7. Prove the Chain Rule using Newton's approximation.
Reading Assignment: Sections 11.1, end of 11.3, 11.4 (the integration laws), 11.5, 11.7 (example of a bounded function that is NOT Riemann integrable), 11.9 (the Fundamental Theorems of Calculus).

Homework 10 (due on Tuesday Nov 28, 2017)

1. Section 9.8: Exercise 9.8.1 (maximum principle for monotone functions).
2. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
3. Section 9.9: Exercise 9.9.12 (uniform continuity preserves Cauchy sequences).
4. Section 10.1: Exercises 10.1.4(g)(h) and 101.6 (quotient rule -use prior rules- and derivative x^n for n<0).
5. Section 10.3: Exercises 10.3.2 and 10.3.3 (Examples involving monotone functions).
6. Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).
7. Bonus: Exercise 9.8.5 (a function that has a discontinuity at each rational number, but is continuous at every irrational number).
Reading Assignment: Sections 9.8-9.9, and Chapter 10.

Homework 9 (due Thursday Nov 9, 2017)

1. Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential defintions of limit of a function at a point).
2. Section 9.3: Exercise 9.3.5 [in Second Edition] (squeeze test).
3. Section 9.4: Exercise 9.4.5 (composition preserves continuity).
4. Section 9.4: Exercise 9.4.7 (polynomials are continuous on R).
5. Section 9.6: Exercise 9.6.1 (examples).
Reading Assignment: Sections 9.1, 9.3-9.6 (9.2 is a review for you on functions).
I will not lecture on Chapter 8 but you are more than welcome to read on your own and ask questions during office hours.

Homework 8 (Due on Thursday Nov 2, 2017)

1. Section 7.1: Exercise 7.1.4 (binomial formula).
2. Section 7.2: Exercise 7.2.3 (Zero test for series or Divergence test).
3. Section 7.2: Exercise 7.2.5. (Series Laws).
4. Section 7.3: Exercise 7.3.1 (Comparison test).
5. Section 7.5: Exercise 7.5.2 (show that a particular series is convergent).
Reading Assignment: Chapter 7.

Homework 7 (due on Thursday Oct 26, 2017)

1. Section 6.1: Exercise 6.1.8 (Limit Laws, recall that being different than zero means being bounded away from zero. Exclude the last two items about min and max you could do them for BONUS points).
2. Given the sequence 1, -1, -1/2, 1, 1/2, 1/3, -1, -1/2, -1/3, -1/4, 1, 1/2, 1/3, 1/4, 1/5, -1, -1/2, -1/3, -14, -1/5, -1/6, 1,1/2,... find its supremum, its infimum, its limsup, its liminf, and all its limit points. Write a short justification for each one of them.
3. Section 6.4: Exercise 6.4.1 (limits are limit points).
4. Section 6.4: Exercises 6.4.5 (squeeze test using comparison principle).
5. Section 6.5: Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
6. Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
Reading Assignment: Chapter 6.

Homework 6 (Due on Thursday Oct 19, 2017, after Fall break):

1. Section 5.4: Exercise 5.4.4 ("Archimedean Principle", use Proposition 5.4.12 or Corollary 5.4.13).
2. Decide if the given statement is TRUE or FALSE. Justify each answer: if true provide a proof, if false a counterexample and a corrected statement.
1. (a) If a nonempty set of R has an upper bound, then it has a least upper bound.
2. (b) If a nonempty set of R has a supremum, then it is bounded.
3. (c) Every nonempty bounded subset of R has a maximum and a minimum.
4. (d) Let S is a nonempty subset of R. If m=inf (S) and m'< m then m' is a lower bound of S.

3. For each subset of R, find, if they exist, its supremum, infimum, maximum, and minimum. Decide whether each set is bounded below, bounded above, or bounded. Explain each answer.
1. (a) The interval I=(0,4].
2. (b) The set A={1/n: n>0 and n is in N}.
3. (c) The set B={r in Q: r^2<=3}.
4. (d) The set C={x in R: x> -5}.
5. (e) The set of integers Z.

4. Let A and B be nonempty bounded subsets of R with a A being a subset of B. Show that sup(A) <= sup(B) and inf(B)<=inf(A).
5. Section 5.5: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise").
6. Bonus Exercise 5.4.3 (interspersing of integers by reals, here Euclidean algorithm will not work). Please write this problem in a separate piece of paper, I will grade it for bonus points).
Reading Assignment: Section 5.4, 5.5 and 5.6.

Homework 5 (due Thursday October 5, 2017)

1. Show that if a sequence of rational numbers converges to a rational number then the sequence is Cauchy. (See Exercise 6.1.5).
2. Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded).
3. Section 5.2: Exercise 5.2.1 (if two sequences are equivalent and one is Cauchy so is the other).
4. Section 5.3: Exercise 5.3.2 (multiplication of real numbers is a well defined real number: 1) product of Cauchy sequences is Cauchy, 2) well defined).
5. Section 5.4: Exercise 5.4.2(c)(d) (transitivity of order, addition does not change order).
Reading Assignment: Chapter 5

Homework 4 (due on Friday Sep 23, 2016):

1. Section 4.1: Exercise 4.1.2 (negation in Z is well defined).
2. Section 4.2: Exercise 4.2.1 (show equality in Q is an honest equality).
3. Section 4.2: Exercise 4.2.6 (show that multiplication by a negative rational number reverses an inequality between two rational number).
4. Section 4.3: Exercise 4.3.2 (d) and (g) (properties of epsilon-close, Proposition 4.3.7, can use previous properties).
5. Section 4.4: Exercise 4.4.1 (Interspersing of integers by rationals).
Note about Exercise 4.2.6: The first edition says reals instead of rationals, clearly a typo. In the second edition the exercise is written correctly.
Reading Assignment: Chapter 4

Homework 3 (due on Thursday Sep 14, 2017):

1. Show that the symmetric difference of two sets is equal to the union of the two sets minus the intersection of the two sets: (A\B)U(B\A)=(AUB)\(A intersect B).
2. Section 3.3: Exercise 3.3.7 (composition of bijections is a bijection moreover you have a formula for the inverse function of the composition).
3. Section 3.5: Exercise 3.5.3 (equality for ordered pairs is an "honest" equality).
4. Section 3.6: Exercise 3.6.5 (cardinality of AxB equals that of BxA, use that together with cardinal arithmetic to show that multiplication in N is commutative).
Reading Assignment: Sections 3.1, 3.3, 3.5, 3.6 (you are more than welcome to also read 3.2 and 3.4).

Homework 2 (due on Thursday Sep 7, 2017):
Knowledge is off. You can use any fact that has been proved before the exercise (including prior exercises in the book), unless the exercise asks you to prove a Theorem/Lemma/Proposition, in that case you can use any fact before the Theorem/Lemma/Proposition. If you are asked to prove item (d) in a proposition, you can use prior items (a), (b), and (c).

1. Section 2.2: Exercise 2.2.2 (existence of a predecessor for positive natural numbers).
2. Not in the book: Let a, b, c, d be in N, if a < b and c< d then a+c < b+d (can use anything in Section 2.2 or before).
3. Section 2.3: Exercise 2.3.2 (N has no zero divisors).
4. Section 2.3: Exercise 2.3.5 (Euclidean Algorithm, show also uniqueness, not just existence).
Reading Assignment: Chapter 2.

Homework 1 (due on Thursday Aug 31, 2017):
For the first two problems turn on your number knowledge, for the last 3 problems turn it off:

1. Show by induction that for all n natural numbers 0+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>=4. (you will need at some point to show that 2n+1<=2^n for n>=4, or to show that n^2>= 2n+1 for n>=4 depending whether you go right to left or left to right in your inequality, that will be an auxiliary lemma that you may prove by induction! in fact these auxiliary inequalities are true for n>=3).
3. Show that for all n in N, 1+n=n++
4. Show that 1+1=2 (hint: use previous exercise!).
5. Show that multiplication is commutative, that is nxm=mxn for all n,m in N (Exercise 2.3.1 in the book).
Hint: prove it by induction on n for each m fixed, and you will need two auxiliary lemmas similar to those used when proving commutativity of addition in class.
Reading Assignment Chapter 2.

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Last updated: Sep 28, 2017