**Homework 11 (last)** (due on Thursday December 1, 2016)

- Section 9.9: Exercise 9.9.3 (uniform continuity preserves Cauchy sequences).

- Section 10.1: Exercise 10.1.4(g)(h) (reciprocal and quotient rule).

- 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 Friday Nov 18th no later than 5 pm)

- Section 9.3: Exercise 9.3.1 (equivalence of epsilon-delta and sequential definitions 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)

- Section 9.8: Exercise 9.8.1 (maximum principle for monotone functions)

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

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

- 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 whether 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 on Friday Oct 28, 2016 no later than 5pm, turned in Steven's office)

- Section 6.1: Exercise 6.1.5 (Convergent sequences are Cauchy sequences).

- Section 6.1: Exercise 6.1.8 (a)(b) (Limit Laws).

- 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.

- 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).

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

- 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 5.5: Exercise 5.5.1 (supremum vs infimum, a "mirror exercise")

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

- 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).

- Section 5.4: Exercise 5.4.2(c)(d) (transitivity of order, addition does not change order).

- Section 5.4: Exercise 5.4.4 ("Archimedean Principle", use Proposition 5.4.12 or Corollary 5.4.13).

- 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).

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

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

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

**Homework 4 **(due on Friday Sep 23, 2016):

- 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) and (g) (properties of epsilon-close, Proposition 4.3.7, can use previous properties).

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

**Homework 3 **(due on Thursday Sep 15, 2016):

- Section 3.1: Exercise 3.1.6(h) (Prove one de Morgan Law: complement of the union is the intersection of the complements).

- 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 3.6.5 (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 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).

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

- 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).

- 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 Thursday Sep 1st, 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.

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Last updated: Aug 24, 2016