MATH 401/501 - HOMEWORK PROBLEMS - Spring 2025

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 6 (Due on Thursday March 27, 2025 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 07, 2025 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 02/26/2025 Lecture 11.
  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 Feb 28, 2025 at 11:59pm):

  1. Given r in Q and n in N, define r^0 :=1 and if r^n in Q then r^{n+1}:=r^n x r. Define $r^{-n}=(r^n)^{-1}$, where r^{-1}=1/r, whenever r is not 0. Show that
    1. r^{-n}=(r^{-1})^n.
    2. 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 Thursday Feb 20, 2025 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. (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.
  6. Bonus: Exercise 4.4.1 (Interspersing of integers by rationals).
Reading Assignment: Chapter 4.

Homework 3 (due on Thursday Feb 13, 2025 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)
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 06, 2025 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. For example, 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) is equivalent to (p or q) and (p or r). Also show that no (p and q) is equivalent to no p or no q.
  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 30, 2025 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.
(Note that a>=b means a is bigger or equal than b, a <= b a is less than or equal to b, for those of you who know LaTeX, respectively \geq and \leq.)

Homework 0 (due on Thursday Jan 23, 2025 at 11:59pm)
This homework you will find in Canvas, is a short exploratory survey.

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Last updated: Jan 19, 2023