### MATH 401/501 - HOMEWORK PROBLEMS - Sring 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 Tuesday May 3, 2016) Do at least 5 of the following exercises.

1. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).
2. Section 9.9: Exercise 9.9.12 (uniform continuity preserves Cauchy sequences).
3. Section 10.1: Exercise 10.1.2 (Newton's Approximation).
4. Section 10.1: Exercise 10.1.4(g)(h) (quotient rule, can assume prior rules and use them).
5. Section 10.1:Exercises 10.1.5 and 10.1.6 (derivative of x^n for n in Z).
6. Section 10.2: Exercise 10.2.5 (Mean Value Thm from Rolle's Thm).
7. Section 10.3: Exercises 10.3.2 and 10.3.3 (Examples involving monotone functions).
8. Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).

Homework 10 (due on Tuesday April 19)

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 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)
Reading Assignment: Sections 9.1, 9.3-9.6 (9.2 is a review for you on functions).

Homework 9 (due on Fr April 1st, 2016): Do 5 problems, of those at least 2 problems from chapter 6 and 2 problems from chapter 7.

1. Section 6.4: Exercise 6.4.1 (limits are limit points).
2. Section 6.4: Exercises 6.4.5 (squeeze test using comparison principle).
3. Section 6.5: Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
4. Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
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 liming and all its limit points. Write a short justification for each one of them.
6. Section 7.1: Exercise 7.1.4 (binomial formula).
7. Section 7.2: Exercise 7.2.2 (Cauchy test for series).
8. Section 7.2: Exercise 7.2.1. (decide wether a series converges or not).
9. Section 7.3: Exercise 7.3.2 (geometric series).
10. Section 7.5: Exercise 7.5.2 (show that a particular series is convergent).
Reading Assignment: Chapters 6 and 7.

Homework 8 (due Fr March 25, 2016)

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 6.1: Exercise 6.1.5 (convergent sequences are Cauchy) [note I did this in class].
5. Section 6.1: Exercise 6.1.8 (a) and (b) (Limit laws).

Homework 7 (due Fr March 11, 2016)

1. Section 5.4: Exercises 5.4.2 (properties (c), (e) of order in R, feel free to use the basic properties of positive reals).
2. Section 5.4: Exercise 5.4.3 (interspersing of integers by reals, you have Proposition 5.4.12 and Archimedean Property 5.4.13. Euclidean Algorithm won't work this time).
3. Section 5.4: Exercise 5.4.4 (one of the Archimedean properties which you can get from others like Property 5.4.13).
4. Section 5.4: Exercise 5.4.6 (properties of absolute value in R).
5. Section 5.5: Exercise 5.5.1(mirror properties of sup and inf: show that a if a set F is not empty and bounded below then it has a greatest lower bound in R, use the LUB property of R on the "mirror set" E=-F).

Homework 6 (due on Friday March 4, 2016, by noon directly delivered to Steven SMLC 246):

1. Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded).
2. Section 5.2: Exercise 5.2.1 (if two sequences are equivalen and one is Cauchy so is the other).
3. Section 5.3: Exercise 5.3.2 (multiplication of reals is well defined).
4. Section 5.3: Exercise 5.3.5 (show that LIM{1/n} = 0 ).

Homework 5 (due on Friday Feb 26, 2016, by noon):

1. Section 4.4: Exercise 4.4.1 (Interspersing of integers by rationals).

Homework 4 (due on Friday Feb 19, 2016, by noon):

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),(f), and (g) (properties of epsilon-close, Proposition 4.3.7).
5. Section 4.3: Exercise 4.3.3 (c)(d) (properties of exponentiation I: rational base, natural exponent. Assume known and use parts (a) and (b) we will discuss them in the review).
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.

Homework 3 (due on Thursday Feb 11, 2016):

1. Section 3.1: Exercise 3.1.10 (show that the 3 sets are "pairwise disjoint" that is the intersection of any two of them will give you the empty set, and show that their union is AUB).
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 3.6.4 (a) and (b) (the first two laws of cardinal arithmetic).
5. 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).
For the last 2 exercises you need to remember what is the cartesian product of two sets AxB := { (a,b): a is in A and b is in B}, we call (a,b) an "ordered pair", so AxB is the collection of ordered pairs with first entry in A and second entry in B. Two ordered pairs are equal: (a,b)=(a',b') iff a=a' AND b=b'. (See Section 3.5 in particular definitions 3.5.1 and 3.5.4)

Homework 2 (due on Thursday Feb 4th, 2016):

1. Section 2.2: Exercise 2.2.2 (existence of a predecessor poor positive natural numbers).
2. Section 2.2: Exercise 2.2.3(e) (a < b iff a++ <= b).
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).

Homework 1 (due on Th Jan 28, 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.