### MATH 401/501 - HOMEWORK PROBLEMS - Fall 2016

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 (last) (due on Thursday December 1, 2016)

1. Section 9.9: Exercise 9.9.3 (uniform continuity preserves Cauchy sequences).
2. Section 10.1: Exercise 10.1.4(g)(h) (reciprocal and quotient rule).
3. Section 10.1: Exercises 10.1.5 and 10.1.6 (derivative of x^n for n in Z).
4. Section 10.2: Exercise 10.2.5 (Mean Value Thm from Rolle's Thm).
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}).
Reading Assignment: Chapter 10.

Homework 10 (due on Friday Nov 18th no later than 5 pm)

1. Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential definitions 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)
6. Section 9.8: Exercise 9.8.1 (maximum principle for monotone functions)
7. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
Reading Assignment: Chapter 9 (Continuous functions on R)

Homework 9 (due on Friday Nov 4th no later than 5 pm)

1. Section 7.1: Exercise 7.1.4 (binomial formula).
2. Section 7.2: Exercise 7.2.2 (Cauchy test for series).
3. Section 7.2: Exercise 7.2.1. (decide whether a series converges or not).
4. Section 7.3: Exercise 7.3.2 (geometric series).
5. Section 7.5: Exercise 7.5.2 (show that a particular series is convergent).
Reading Assignment: Chapter 7 (Series)

Homework 8 (due on Friday Oct 28, 2016 no later than 5pm, turned in Steven's office)

1. Section 6.1: Exercise 6.1.5 (Convergent sequences are Cauchy sequences).
2. Section 6.1: Exercise 6.1.8 (a)(b) (Limit Laws).
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 6.4.5 (squeeze test using comparison principle).
6. Section 6.5: Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
7. Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
Reading Assignment: Chapter 6 (Limits of sequences)

Homework 7 (Due on Friday October 21 no later than 5pm, after Fall break):

1. Decide if the given statement is TRUE or FALSE. Justify each answer: if true provide a short proof, if false provide a counterexample and a corrected statement.
1. If a nonempty set of R has an upper bound, then it has a least upper bound.
2. If a nonempty subset of R has an supremum, then it is bounded.
3. Every nonempty bounded subset of R has a maximum and a minimum.
4. If m=inf S and m'< m, then m' is a lower bound for S.
2. For each subset of R, give, if they exist its supremum, infimum, maximum, and minimum. Decide whether the set is bounded above, bounded below, or bounded. Explain your answers.
1. the interval (0,4],
2. the set {1/n: n>0 is in N}={1,1/2,1/3,1/4,1/5,...},
3. the set {r in Q: r^2 <= 3},
4. the set {x in R: x > -5},
5. the set of integers Z.
3. Let A and B be nonempty bounded subsets of R with A a subset of B. Show that sup(A) <= sup(B).
4. Section 5.5: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise")
Reading Assignment: Chapter 5 (Sections 5 and 6)

Homework 6 (due on Wednesday Oct 12, 2016):

1. Section 5.3: Exercise 5.3.2 (multiplication of reals produces a well defined real number: 1) product of Cauchy is Cauchy, 2) well defined).
2. Section 5.4: Exercise 5.4.2(c)(d) (transitivity of order, addition does not change order).
3. Section 5.4: Exercise 5.4.4 ("Archimedean Principle", use Proposition 5.4.12 or Corollary 5.4.13).
4. Section 5.4: 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: Chapter 5 (Sections 3, 4).

Homework 5 (due on Thursday Oct 6, 2016):

1. Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded).
2. Section 5.2: Exercise 5.2.1 (if two sequences are equivalent and one is Cauchy so is the other).
Reading Assignment: Chapter 5 (Sections 1,2 and 3).

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 15, 2016):

1. Section 3.1: Exercise 3.1.6(h) (Prove one de Morgan Law: complement of the union is the intersection of the complements).
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 ready 3.2 and 3.4).

Homework 2 (due on Thursday Sep 8st, 2016):
Knowledge is off. You can use any fact that has been proved before the exercise, 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. Section 2.2: Exercise 2.2.3(d) (addition preserves order, note this is an if and only if statement, it means you have to prove two implications, the direct implication and the converse implication).
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 Sep 1st, 2016):
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 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 n<=2^n for all n, that will be an auxiliary lemma that you may prove by induction!).
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 we will need when proving commutativity of addition on Tuesday.

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Last updated: Aug 24, 2016