MATH 401/501 - HOMEWORK PROBLEMS - Spring 2023

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 10 (due on Friday April 28, 2023, at 11:59pm)

  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 10.3: Exercises 10.3.2 (Example involving a monotone function).
  4. Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).
  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? Assume now that f and g are Riemann integrable on [a,b] and show that (f+g) is Riemann integrable on [a,b], moreover int_[a,b] (f+g) = int_[a,b] f+int_[a,b] g (Exercise 9.4.1 (a)).
  7. Bonus: Exercise 10.1.7 (prove the Chain Rule using Newton's approximation).
Reading Assignment: Chapter 10 (derivatives), Chapter 11 (Riemann integration) focus on 11.1, end of 11.3 (starting at Definition 11.3.9 of Riemann 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 9 (due Friday April 21, 2023)

  1. Section 9.4: Exercise 9.4.5 (composition preserves continuity, use epsilon-delta argument).
  2. Section 9.4: Exercise 9.4.7 (polynomials are continuous on R).
  3. Section 9.6: Exercise 9.6.1 (examples).
  4. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
  5. Section 9.9: Exercise 9.9.3 (uniform continuity preserves Cauchy sequences).
  6. 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).
  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: Read Sections 9.6-9.9 and 10.1.

Homework 8 (Due on Friday April 07, 2023 at 11:59pm)

  1. Section 7.3: Exercise 7.3.1 (Comparison test).
  2. Section 7.5: Exercise 7.5.2 (a specific family of series).
  3. Section 7.5: Exercise 7.5.3 (examples of series in the inconclusive regime for Math 163 root and ratio test).
  4. Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential definitions of limit of a function at a point).
  5. Let f,g be functions defined from R into R, and let a,b be real numbers. Show that if f and g are continuous at x_0 in R then (af+bg) is continuous at x_0.
Reading Assignment: Sections 7.3-7.5, Sections 7.1 (on finite sums) is to be read by you. We are skipping Chapter 8, happy to talk about it during office hours or after class. Sections 9.1-9.4, Section 9.2 is to be read by you.

Homework 7 (Due on Friday March 31st, 2023 at 11:59pm)

  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 6.6.4 (subsequences related to limits, see remark 6.6.7).
  4. Section 7.2: Exercise 7.2.3 (zero test for series or Divergence test).
  5. Section 7.2: Exercise 7.2.5. (Series Laws).
  6. Bonus: Exercise 6.1.8 (g) (limit law involving max).
  7. Bonus Exercise 7.1.4 (binomial formula).
Reading Assignment: Sections 6.4, 6.5, 6.6 and 6.7 in Chapter 6 and Chapter 7.

Homework 6 (Due on Friday March 24, 2023 at 11:59pm):

  1. 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 subset of R has an upper bound, then it has a least upper bound.
    2. (b) If a nonempty subset 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.
  2. 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.
  3. Let A and B be nonempty bounded subsets of R with A being a subset of B. Show that sup(A) <= sup(B) and inf(B) <= inf(A).
  4. 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).
  5. 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.
  6. Bonus: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise").
  7. Bonus: Section 6.4: Exercise 6.4.10 (limit-points of limit-points are themselves limit-points of the original sequence).
Reading Assignment: Chapter 6

Homework 5 (due Friday March 10, 2023 at 11:59pm)

  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: (i) product of Cauchy sequences is Cauchy, (ii) well defined).
  5. Negate mathematical statements involving quantifiers: (i) a sequence is not Cauchy, (ii) two sequences are not equivalent, (iii) a sequence is not convergent to L, (iv) a sequence is not a bounded. More details on what is expected on the class summary for 02/28/2023 Lecture 10.
  6. 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
  7. Bonus Exercise 5.4.4 ("Archimedean Principle", use Corollary 5.4.13).
  8. Bonus Exercise 5.4.3 ("Interspersing of integers by reals", here Euclidean algorithm won't work).
Reading Assignment: Chapter 5

Bonus Homework (due on Friday March 3, 2023 at 11:59pm):

  1. (Archimedean properties in Q) Given rational numbers x, y, z, e. Show that the following three statements are equivalent:
    1. Given x>0 there is p in N such that x < p.
    2. Given y>0 and e>0 there is m in N such that y < em.
    3. Given z>0 there is n in N such that (1/n) < z.
    To show the three statements are equivalent it seems like you need to show 6 implications (1) <=> (2), (2) <=> (3), and (3)<=>(1), however you can get away proving only 3 implications, for example: (1)=>(2)=>(3)=>(1) or (3) =>(1)=>(2)=>(3) or several other combinations. (Once you know they are equivalent proving one of them proves them all. Note that we have already proved (1) [in homework 4 as a consequence of the interspersing of integers and rationals], (2) was in the review for the exam we showed it as a consequence of (1), finally (3) was in your exam. )
  2. Given r in Q and n in N, define r^0 :=1 and if rn in Q then r^{n+1}:=r^n x r. In your exam you showed that if n,m in N then r^{n+m}=r^n x r^m. If r is not 0, define r^{-n}:=1/(r^n)=(r^n)^{-1}. Now show
    1. If a,b in Z then r^{a+b}=r^a x r^b.
    2. Show that r^{-n}=(r^{-1})^n.
    3. If r, s in Q and a in Z then (r x s)^a=r^a x s^a. (You may want to first consider the case a=n in N, then the case a=-n).

Homework 4 (due on Friday Feb 17, 2023 at 11:59pm):

  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. Bonus: 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 Friday Feb 10, 2023 at 11:59pm):

  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 3.3, 3.5, 3.6 (you are more than welcome to also read 3.4).

Homework 2 (due on Thursday Feb 2nd, 2023 at 11:59pm):
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 (Euclid's Division Lemma, in prior editions called 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)
  6. Bonus: Prove Proposition 2.2.12(e): if a< b then a++<=b.
Reading Assignment: Sections 2.3, 3.1 (sets) and 3.3 (functions) (you are more than welcome to read also Section 3.2 (Russel's Paradox) I will not lecture on this however).

Homework 1 (due on Thursday Jan 26, 2023 at 11:59pm)
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, n++=1+n=n+1 using only the definition of addition not the commutative property. Then show that 1+1=2.
  4. Show that multiplication in N 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.

Homework 0 (due on Thursday Jan 19, 2023 at 11:59pm)
This homework you will find in Canvas, is a short exploratory survey.

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Last updated: Jan 19, 2023