MATH 401/501 - HOMEWORK PROBLEMS - Spring 2020

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 (due on Monday May 4th, 2020)

  1. Section 10.1: Exercises 10.1.4(g)(h) and 10.1.6 (quotient rule -use prior rules- and derivative x^n for n<0).
  2. Section 10.2: Exercises 10.2.2 (Example of function attaining local max at 0 but not differentiable at 0).
  3. Section 10.2: Exercise 10.2.5 (Mean Value Theorem from Rolle's Theorem).
  4. Section 10.2: Exercises 10.2.6 and 10.2.7 (functions with bounded derivative are Lipschitz and are uniformly continuous).
  5. Section 10.3: Exercises 10.3.2 (Example involving a monotone function).
  6. Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).
  7. Bonus: Exercise 10.1.7 (prove the Chain Rule using Newton's approxi\ mation).
  8. Bonus: Exercise 9.8.5 (a function that has a discontinuity at each rational number, but is continuous at every irrational number).
Reading Assignment: Chapter 10 (derivatives)

Homework 10 (due Monday April 20, 2019)

  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 and Third Edition] (continuous of the 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).
  6. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
  7. Section 9.9: Exercise 9.9.3 (uniform continuity preserves Cauchy sequenc\ es).
Reading Assignment: Read Chapter 9. I won't directly lecture on Section 9.1 but introduce definitions as necessary, Section 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 9 (Due on Monday April 13, 2020)

  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).<\ BR>
  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 (a specific family of series).
  6. Section 7.5: Exercise 7.5.3 (examples of series in the inconclusive regim\ e for Math 163 root and ratio test).
Reading Assignment: Chapter 7. Sections 7.1 (on finite sums) and 7.4 (rearrangement of series) are to be read by you.

Homework 8 (Due on Wednesday April 8, 2020)

  1. Section 6.4: Exercises 6.4.4 and 6.4.5 (comparison principles and squeeze test).
  2. Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
  3. Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
  4. Section 6.6: Exercise 6.6.4 (subsequences related to limits, see remark 6.6.7).
Reading Assignment: Sections 6.4, 6.5, 6.6 and 6.7.

Homework 7 (due on Sunday March 29, 2020)

  1. Section 5.5: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise").
  2. Section 6.1: Exercise 6.1.8(a)(b)(f) (Limit Laws, recall that the limit being different than zero means the sequence is eventually bounded away from zero).
  3. 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.
  4. Section 6.4: Exercise 6.4.1 (limits are limit-points).
  5. Section 6.4: Exercises 6.4.7 (zero test for sequences, do this using the definition of convergent sequences).
  6. Section 6.4: Exercise 6.4.10 (limit-points of limit-points are themselves limit-points of the original sequence).
  7. Bonus: Exercise 6.1.8 (g) or (h) (limit laws involving max or min).
Reading Assignment: Chapter 6.

Homework 6 (Due on Thursday March 12, 2020):

  1. Show that for all x,y,z in R (use the definition of real numbers as Cauchy sequences in Q and that the corresponding properties are true in Q)
    1. (a) 1.x = x
    2. (b) y-y = 0
    3. (c) If z is not 0 then z.(1/z) = 1
    4. (d) (x+y)z = x.z + y.z
    5. (e) if x < y then x+z < y+z
  2. Section 5.4: Exercise 5.4.4 ("Archimedean Principle", use Corollary 5.4.13).
  3. 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 an infimum, then it is bounded.
    3. (c) Every nonempty bounded subset of R has a maximum and a minimum.
    4. (d) Let S be a nonempty subset of R. If m=inf (S) and m'< m then m' is a lower bound of S.
  4. 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.
  5. 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).
  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 March 5, 2020)

  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. Negate mathematical statements involving quantifiers: not Cauchy, not equivalent sequences, not convergent to L, not a bounded sequence). More details on what is expected on the class summary for 2/27/20 Lecture 12.
Reading Assignment: Chapter 5

Homework 4 (due on Thursday Feb 20, 2020): Please write the bonus in a separated piece of paper and don't staple it to the homework.

  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).
  6. Bonus: Exercise 3.6.4 (Cardinal arithmetic)
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 Feb 13, 2020):

  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 3.6: Exercise 3.6.1 (equal cardinality 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).
  5. Bonus: Exercise 3.6.6 (about power sets) -I will grade it, please turn it in on a separate piece of paper.
Reading Assignment: Sections 3.5, 3.6 (you are more than welcome to also read 3.4).

Homework 2 (due on Thursday Jan 31st, 2019):
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).
  5. Not in the book: use truth tables to prove the distributive property for OR and AND, that is if p, q, r are statements then p or (q and r) <=> (p or q) and (p or r)
Reading Assignment: Sections 2.3, 3.1 (sets) and 3.3 (functions) (you are more welcome to read also Section 3.2 (Russel Paradox) I will not lecture on this however).

Homework 1 (due on Thursday Jan 30, 2020):
For the first two problems turn on your number knowledge, for the last two 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++ . Then show that 1+1=2.
  4. 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: Feb 28, 2020