** Homework 11 **(last) (due on Tuesday May 3, 2016) Do at least 5 of the following exercises.

- Section 9.8: Exercise 9.8.2 (Intermediate Value Theorem example).

- Section 9.9: Exercise 9.9.12 (uniform continuity preserves Cauchy sequences).

- Section 10.1: Exercise 10.1.2 (Newton's Approximation).

- Section 10.1: Exercise 10.1.4(g)(h) (quotient rule, can assume prior rules and use them).
- Section 10.1:Exercises 10.1.5 and 10.1.6 (derivative of x^n for n in Z).

- Section 10.2: Exercise 10.2.5 (Mean Value Thm from Rolle's Thm).

- Section 10.3: Exercises 10.3.2 and 10.3.3 (Examples involving monotone functions).

- Section 10.4: Exercise 10.4.1 (derivative of x^{1/n}).

** Homework 10** (due on Tuesday April 19)

- Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential defintions of limit of a function at a point).

- Section 9.3: Exercise 9.3.5 [in Second Edition] (squeeze test)
- Section 9.4: Exercise 9.4.5 (composition preserves continuity)
- Section 9.4: Exercise 9.4.7 (polynomials are continuous on R)
- Section 9.6: Exercise 9.6.1 (examples)

** 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.

- Section 6.4: Exercise 6.4.1 (limits are limit points).
- Section 6.4: Exercises 6.4.5 (squeeze test using comparison principle).
- Section 6.5: Exercise 6.5.3 (limit of n-th root of x>0 as n goes to infinity is one).
- Section 6.6: Exercise 6.6.2 (create two different sequences so that each is a subsequence of the other).
- 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.
- Section 7.1: Exercise 7.1.4 (binomial formula).
- Section 7.2: Exercise 7.2.2 (Cauchy test for series).
- Section 7.2: Exercise 7.2.1. (decide wether a series converges or not).
- Section 7.3: Exercise 7.3.2 (geometric series).
- Section 7.5: Exercise 7.5.2 (show that a particular series is convergent).

**
Homework 8** (due Fr March 25, 2016)

- 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.
- If a nonempty set of R has an upper bound, then it has a least upper bound.
- If a nonempty subset of R has an supremum, then it is bounded.
- Every nonempty bounded subset of R has a maximum and a minimum.
- If m=inf S and m'< m, then m' is a lower bound for S.

- 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.
- the interval (0,4],
- the set {1/n: n>0 is in N}={1,1/2,1/3,1/4,1/5,...},
- the set {r in Q: r^2 <= 3},
- the set {x in R: x > -5},
- the set of integers Z.

- Let A and B be nonempty bounded subsets of R with A a subset of B. Show that sup(A) <= sup(B).
- Section 6.1: Exercise 6.1.5 (convergent sequences are Cauchy) [note I did this in class].
- Section 6.1: Exercise 6.1.8 (a) and (b) (Limit laws).

**
Homework 7** (due Fr March 11, 2016)

- Section 5.4: Exercises 5.4.2 (properties (c), (e) of order in R, feel free to use the basic properties of positive reals).

- 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).

- Section 5.4: Exercise 5.4.4 (one of the Archimedean properties which you can get from others like Property 5.4.13).

- Section 5.4: Exercise 5.4.6 (properties of absolute value in R).

- 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):

- Section 5.1: Exercise 5.1.1 (Cauchy sequences are bounded).

- Section 5.2: Exercise 5.2.1 (if two sequences are equivalen and one is Cauchy so is the other).

- Section 5.3: Exercise 5.3.2 (multiplication of reals is well defined).

- Section 5.3: Exercise 5.3.5 (show that LIM{1/n} = 0 ).

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

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

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

- Section 4.1: Exercise 4.1.2 (negation in Z is well defined).

- Section 4.2: Exercise 4.2.1 (show equality in Q is an honest equality).

- Section 4.2: Exercise 4.2.6 (show that multiplication by a
negative rational number reverses an inequality between two rational number).

- Section 4.3: Exercise 4.3.2 (d),(f), and (g) (properties of epsilon-close, Proposition 4.3.7).

- 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).

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

- 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).

- 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).

- Section 3.5: Exercise 3.5.3 (equality for ordered pairs is an "honest" equality).

- Section 3.6: Exercise 3.6.4 (a) and (b) (the first two laws of cardinal arithmetic).

- 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).

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

- Section 2.2: Exercise 2.2.2 (existence of a predecessor poor positive natural numbers).

- Section 2.2: Exercise 2.2.3(e) (a < b iff a++ <= b).

- Section 2.3: Exercise 2.3.2 (N has no zero divisors).

- 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:

- Show by induction that for all n natural numbers
1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

- 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!).

- Show that for all n in N, 1+n=n++

- Show that 1+1=2 (hint: use previous exercise!).

- 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.

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: Feb 23, 2016