### MATH 401/501 - HOMEWORK PROBLEMS - Fall 2014

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 12 (last) (due on Wed Nov 26, 2014, you get 5 bonus points if you turn it in on Tuesday Nov 25, 2014).

1. Section 10.1: Exercise 10.1.2 (Newton's Approximation),
Exercises 10.1.5 and 10.1.6 (derivative of x^n for n in Z).
2. Section 10.2: Exercise 10.2.5 (Mean Value Thm from Rolle's Thm).
3. Section 10.3: Exercises 103.2 and 10.3.3 (Examples involving monotone fcns).
4. Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).

Homework 11 (due on Wednesday Nov 19, 2014, if turned in on Tue Nov 18 you get 5 bonus points):

1. Section 9.4: Exercise 9.4.5 (composition preserves continuity)
2. Section 9.5: Exercise 9.5.1 (define a limit to be infinity)
3. Section 9.6: Exercise 9.6.1 (examples)
4. Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example)
5. BONUS: Exercise 9.8.5
Reading Assignment: Sections 9.5, 9.6, 9.7, 9.8

Homework 10 (due on Wednesday Nov 12, 2014, if turned in on Tue Nov 11, you get bonus points):

1. Section 7.5: Exercises 7.5.2 (if |x|<1, q in Q, then sum_{n>=1}n^qx^n is absolutely convergent).
Exercise 7.5.3 (ratio and root tests are inconclusive in "can't tell" regime).
BONUS: Exercise 7.5.1
2. Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential defintions of limit of a function at a point).
Exercise 9.3.5 [in Second Edition] (squeeze test).

Homework 9 (due on Th Oct 30, 2014, if turned in on Tue Oct 28 you get 5 bonus points)

1. Section 6.4: Exercise 6.4.4 (Comparison principles for sup, inf, liminf and limsup. Note that you get the later two from the first ones).
Exercise 6.4.5 (squeeze test)
Exercise 6.4.9 (a sequence with three limit points)
2. Section 6.5: Exercise 6.5.2 (when does {x^n} converge? )
3. Section 6.6: Exercise 6.6.2 (sequences that are subsequences of each other)
Reading Assignment: Sections 6.4, 6.5, 6.6 and 6.7.

Homework 8 (due Th Oct 23, 2014, you get 5 bonus points if turned in on Tu Oct 21, 2014)

1. Section 5.5: Exercise 5.5.1 (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).
2. Section 5.6: Exercise 5.6.1(a)(b)(e) (properties of nth roots)
3. Section 6.1: Exercise 6.1.5 (show that convergent sequences in R are Cauchy in R), Exercise 6.1.6 (Show that formal limits are genuine limits), Exercise 6.1.8 (a)(e) and BONUS: (g) (Limit laws).
Reading assignment: Sections 5.5, 5.6 and 6.1

Homework 7 (due Th Oct 16, 2014, you get 5 bonus points if turned in on Tu Oct 14, 2014)

1. Section 5.4: Exercises 5.4.2 (properties of order in R)
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, but the algorithm I described as an alternative proof when we did this for Q will work for positive reals).
2. Exercise 5.4.4 (one of the Archimedean properties which you can get from others like Property 5.4.13)
3. Exercise 5.4.6 (properties of absolute value in R).
4. BONUS: Exercise 5.4.1 (except for trichotomy, which you can assume known).

Homework 6 (due on Wed October 8th, 2014, you get 5 bonus points if you turn it in on Tu October 7th, 2014).

1. Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded, Erik gave you a hint last Wednesday is recorded in his notes)
2. Section 5.2: Exercise 5.2.1 (if two sequences are equivalent and one is Cauchy so is the other).
3. Section 5.3: Exercise 5.3.2 (multiplication of reals is well defined).
Exercise 5.3.4 (show that LIM{1/n}=0).
Reading assignment: Sections 5.1, 5.2 and 5.3.

Homework 5 (due Th Sep 25, if turned in on Tu Sep 23 you get 5 bonus points):

1. Section 4.3: Exercise 4.3.2 (d)(f) and (g) (properties of epsilon-close, Proposition 4.3.7),
Exercise 4.3.3 (c)(d) (properties of exponentiation I: rational base, natural exponent).
2. Section 4.4: Exercise 4.4.1 (interpersing integers by rationals).
3. BONUS: Exercise 4.4.2 (principle of infinite descent).

Homework 4 (due on Th Sep 18, 2014, or Tue Sep 16 for 5 bonus points):

1. Section 4.1: Exercise 4.1.2 (negation in Z is well defined)
Exercise 4.1.5 (integers have no zero divisors)
2. Section 4.2: Exercise 4.2.1 (show equality in Q is an honest equality)
Exercise 4.2.6 (show that multiplication by a negative rational number reverses an inequality between two rational numbers)
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 Th Sep 11, 2014, if turned in on Tue Sep 9 you get 5 bonus points. You get bonus points for the bonus problems even if you turn them in on Thursday):

1. Section 3.1: Exercise 3.1.10 (show that A\B, B\A and A intersect B are "pairwise disjoint" -take any two and their intersection is the empty set- and show their union is AUB).
2. Section 3.3: Exercise 3.3.7 (about invertibility of the composition of two invertible functions).
3. Section 3.5: Exercise 3.5.3 (show equality for ordered pairs is an honest equality).
4. Section 3.6: Exercise 3.6.5 (show #(AxB)=#(BxA), use that together with cardinal arithmetic to show that multiplication in N is commutative).
5. BONUS: Exercises 3.6.4 (cardinal arithmetic) or Exercise 3.6.6. (please turn this in a separate sheet of paper since I will grade it).
For the last non bonus 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} "ordered pairs". 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 Th Sep 5, 2014, if turned in on Tue Sep 3 you get 5 bonus points):

1. Section 2.2: Exercise 2.2.2 (existence of a predecessor for 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 uniqueness not just existence).

Homework 1 (due on Th Aug 28, 2014, if turned in on Tue Aug 26 you get 5 bonus points):
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 like we will need when proving commutativity of addition on Tuesday.