MATH 401/501 - HOMEWORK PROBLEMS - Spring 2019

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 Thursday April 25, 2019). Do four of the following problems, the rest will count 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 10.4: Exercise 10.4.1 (derivative of x^{1/n}).
  4. Section 10.1: Exercise 10.1.7 (prove the Chain Rule using Newton's approximation).
  5. Section 11.1: Exercise 11.1.4 (on common refinement of partitions).
  6. 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?
  7. 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)).
  8. Section 11.9: Exercise 11.9.2 (antiderivative for the same function differ by a constant).
Reading Assignment: Sections 10.4 (derivative of inverse functions), 11.1, end of 11.3 (starting at Definition 11.3.9 of Rieman sums), 11.4 (the integration laws no proofs), 11.5 (continuous functions are Riemann integrable), 11.7 (example of a bounded function that is NOT Riemann integrable), 11.9 (the Fundamental Theorems of Calculus).

Homework 10 (due on Thursday April 18, 2019)

  1. Section 9.6: Exercise 9.6.1 (examples).
  2. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
  3. Section 9.9: Exercise 9.9.3 (uniform continuity preserves Cauchy sequences).
  4. 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).
  5. Section 10.2: Exercises 10.2.2 (Example of function attaining local max at 0 but not differentiable at 0).
  6. Section 10.3: Exercises 10.3.2 (Example involving a monotone function).
  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 9.6-9.9, and Sections 10.1-10.3.

Homework 9 (due Thursday April 4, 2019)

  1. Section 7.5: Exercise 7.5.2 (a specific family of series).
  2. Section 7.5: Exercise 7.5.3 (examples of series in the inconclusive regime for 163 root and ratio test).
  3. Section 9.3: Exercise 9.3.5 [in Second Edition] (squeeze test).
  4. Section 9.4: Exercise 9.4.5 (composition preserves continuity).
  5. Section 9.4: Exercise 9.4.7 (polynomials are continuous on R).
  6. Bonus: Exercise 9.3.1 (equivalence of epsilon-delta and sequential defintions of limit of a function at a point).
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 March 28, 2019)

  1. Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
  2. Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
  3. Section 7.2: Exercise 7.2.3 (Zero test for series or Divergence test).
  4. Section 7.2: Exercise 7.2.5. (Series Laws).
  5. Section 7.3: Exercise 7.3.1 (Comparison test).
  6. Bonus Problem: Exercise 7.1.4 (binomial formula).
Reading Assignment: Chapter 7.

Homework 7 (due on Thursday March 21, 2019)

  1. 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).
  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 6.4.4 (comparison principles).
  5. Section 6.4: Exercises 6.4.7 (zero test for sequences).
  6. Bonus: Exercise 6.1.8 (g) or (h) (limit laws involving max or min), Exercise 6.4.10 (limit-points of limit-points are themselves limit-points of the original sequence).
Reading Assignment: Chapter 6.

Homework 6 (Due on Thursday March 7, 2019):

  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
  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 a infimum, 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.
  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. Section 5.5: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise").
  7. 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 Feb 28, 2019)

  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. Bonus 5 points described at the end of the message on lecture 10 (negating mathematical statements involving quantifiers: not Cauchy, not equivalent sequences, not convergent to L, not a bounded sequence) please turn in separately.
Reading Assignment: Chapter 5

Homework 4 (due on Friday Feb 14, 2019): 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)
  7. Bonus: if f:X-->Y and g:Y--Z then gof:X-->Z. Assume gof is injective, what about f and g? Assume now gof is surjective, what about f and g? What if gof is a bijection? See Exercise 3.3.5.
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 7, 2019):

  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).
  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 24, 2019):
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: April 18, 2019