** Homework 11 (last)** (due on Thursday May 1st, if you turn it in on
Tuesday April 29 you get 5 bonus points).

- Section 10.1: Exercises 10.1.4(g), 10.1.5.
- Section 10.2: Exercise 10.2.2 (Practice 10.2.3)
- Section 10.3: Exercise 10.3.3 (Practice 10.3.2)
- Section 10.4: Exercise 10.4.1.

**BONUS HOMEWORK** (due anytime before the end of Review week on May 8th)

- Section 4: Exercise 9.4.7
- Section 5: Exercise 9.5.1
- Section 7: Exercise 9.7.2
- Section 8: Exercise 9.8.4.

** Homework 10** (due on Thursday April 24 (if turned in on
Tuesday April 22, 5 bonus points)

- Section 9.3: Exercise 9.3.1 (Practice 9.3.2, 9.3.5)
- Section 9.4: Exercise 9.4.2 (examples) (Practice 9.4.1, 9.4.3, 9.4.4)
- Section 9.6: Exercise 9.6.1 (examples)
- Section 9.8: Exercise 9.8.2 (examples) (Practice 9.8.1) (BONUS: 9.8.5)
- Section 9.9: Exercise 9.9.6 (composition preserves uniform continuity)

** Homework 9** (due Thursday April 17 - you get 5 bonus points if turned in
on Tuesday April 15):

- Section 7.5: Exercises 7.5.1 (an inequality comparing liminf of ratio
and nth root), 7.5.2 (a specific power series) and 7.5.3 (these are
examples).

- Section 9.1: Exercise 9.1.3 (for this you need Definition 9.1.10 of the
Closure of a Set: the collection of all adherent points to E; and you
need Definition 9.1.8 of adherent points to a set: x in R is an
adherent point to a set E in R iff for all epsilon >0 there is y in E
such that |y-x|< epsilon.

- Section 9.2: Exercise 9.2.1 (about real valued functions defined on R).
- Bonus: Exercise 7.4.1

- Reading Assignment: Sections 7.4-7.5 and Sections 9.1-9.2

** Homework 8** (due Thursday April 3 - you get 5 bonus points if turned in
on Tuesday April 1):

- Section 6.3: Exercise 6.3.1 (Practice 6.3.4).

- Section 6.4: Exercises 6.4.5 (squeeze test), 6.4.6 and 6.4.9 (build
examples) (Practice Exercise 6.4.4, 6.4.9, BONUS 6.4.10.

- Section 6.5: Exercise 6.5.2 (Practice 6.5.1).

- Section 6.6: Exercises 6.6.2 (build an example), 6.6.4 (subsequences vs
limits).

- Section 6.7: BONUS Exercise 6.7.1

**Homework 7** (due Thursday 3/27/14, 5 bonus points if you turn this in
on Tuesday 3/25/14):

- Section 5.5: Exercise 5.5.1

- Section 5.6: Exercise 5.6.2(a)(c) (Practice 5.6.1, BONUS: 5.6.2(b)(d)(e))

- Section 6.1: Exercise 6.1.5 (convergent in R implies Cauchy in R),

Exercise 6.1.8 (b) (e) (limit laws) (BONUS: 6.1.6, 6.1.8(g)) [Practice: 6.1.4, 6.1.10]

**Homework 6** (due Thursday 3/13/14, 5 bonus points if you turn this in
on Tuesday 3/11/14):

- Section 5.3: Exercises 5.3.2 (Multiplication is well defined, I did half of this exercise in class), 5.3.5 (LIM{1/n}=0) (Practice 5.3.3)

- Section 5.4: Exercise 5.4.4 (an Archimedean Principle) (BONUS: 5.4.3 and 5.4.7) (Practice: 5.4.2, 5.4.5)

**Homework 5** (due Thursday 3/6/14, 5 bonus points if you turn this in
on Tuesday 3/4/14):

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

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

**
Homework 4 ** (due Thursday 2/20/14, 5 bonus points if you turn this in
on Tuesday 2/18/14):

- Section 4.1: Exercises 4.1.2 and 4.1.7(d) (Practice 4.1.3, 4.1.4, 4.1.8)

- Section 4.2: Exercise 4.2.6 (Practice 4.2.2, 4.2.7)

- Section 4.3: Exercise 4.3.2(f)(g)

- Section 4.4: BONUS Exercise 4.4.1

** Homework 3 ** (due Th Feb 13, 2014, if turned on Tue Feb 11 get 5 bonus points):

- Section 3.1: Exercise 3.1.10 (Practice: 3.1.6 and 3.1.8))

- Section 3.3: Exercise 3.3.7 (Practice 3.3.2 and 3.3.5))

- Section 3.4: Exercise 3.4.2 (Practice 3.4.1 and 3.4.5))

- Section 3.5: Exercise 3.5.3 (Practice 3.5.4))

- Section 3.6: Exercise 3.6.5 (BONUS: 3.6.9 or 3.6.6 to be turned in
separately to me)

**Homework 2 ** (due Th Feb 6, 2014, if turned on Tue Feb 4 get 5 bonus points):

- Exercise 2.3.1 (p.32 2nd ed, p.36 1st ed) (multiplication is commutative)

- Exercise 2.2.3 (d) OR (f) [p.29 2nd ed, p.33 1st ed] (Properties of order)

- Exercise 2.3.5 [p.32 2nd ed, p.36 1st ed] (Euclidean Algorithm). Also
show that the m,r you find are unique.

**Homework 1 ** (due on Th Jan 30, 2014, if turned in on Tue Jan 28 get 5 bonus points):

For the first two problems assume arithmetic and whatever
you know about inequalities (turn on your number knowledge):

- 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>3.

- Show that n+0 = 0+n (here you turn off your knowledge, all you know is
the Peano axioms and the definition of addition).

Hint: prove it by induction.

** Quiz 1** solved and more.

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

Last updated: January 23, 2014