MATH 361/461 - HOMEWORK PROBLEMS - Fall 2001

Text: Analysis with an introduction to proofs, by S. Lay

Homework 1 (due Aug 28)

Section 1: 1.2, 1.4, 1.5, 1.8, 1.10, 1.12 (a) (c) (e).
Section 2: 2.2, 2.4, 2.6 (a) (b), 2.13, 2.14, 2.15.

Homework 2 (due Sep 4)

Section 3: 3.2, 3.3, 3.4, 3.5, 3.6 (b) (e) (f) (g), 3.7, 3.8.
Section 4: 4.2, 4.3, 4.6, 4.8, 4.10, 4.11, 4.14.

Homework 3 (due Sep 11 or Sep 13 if need be)

Section 5: 5.2, 5.3 (c) (d) (e) (g) (h), 5.4 (c) (f) (g), 5.6, 5.8, 5.10 (a) (c) (e), 5.12, 5.14, 5.18 (a) (c), 5.19.
Section 6: 6.2, 6.4, 6.5 (a) (c) (e) (f) (h), [think about 6.6], 6.8, 6.10, 6.12, 6.14.

Homework 4 (due Sep 18)

Section 7: [Think about 7.1 and 7.2], 7.3 (b) (d), 7.4, 7.6, [think about 7.8], 7.9, 7.10, 7.14, 7.16, 7.17, 7.18, 7.22.

Homework 5 (due Sep 25)

Section 8: 8.2, 8.3, 8.5, [Think about 8.16, 8.18].
Section 10: 10.3, 10.4, 10.6, 10.7 (use induction!), 10.8, 10.10, 10.11, 10.13 (b) (c), 10.14, [Think about 10.15], 10.18.

Homework 6 (due Oct 9)

Section 11: 11.6, 11.7 (triangle inequalities, see Theorem 11.9), 11.9 (define ordered field via positive elements).
Section 12: 12.3 (f) (h) (i) (n), 12.4 (f) (h) (i) (n), 12.7, 12.12.

Project (due Oct 9)

Problem 11.12 - Construction of rationals from integers.
Problem 11.13 - Construction of integers from natural numbers.
(Hint: define the equivalence relation on NxN by (n,m)R(p,q) iff n+q=m+p ).
Complementary reading: Problem 10.21 - on the construction of the natural numbers from Peano axioms.

Homework 7 (due Oct 17)

Section 13: [think 13.1, 13.2], 13.3, 13.4, 13.5, 13.10, 13.11, 13.16 (c) (d), 13.18, 13.20.
Section 14: [think 14.1, 14.2], 14.3, 14.4, 14.11 (bonus), 14.12.

Homework 8 (due Nov 1)

Section 16: [Think: 16.1 (c) (d), 16.2, 16.3], [Examples: 16.4 (b) (c) (d), 16.5 (e) (f), 16.6 (c) (d)], 16.7, 16.10, 16.11 (sandwich theorem), [Connections to topology: 16.13, 16.14].
Section 17: [Think: 17.1 (b) (d), 17.2 (c) (d)], [Examples: 17.4 (b), 17.5 (c) (d) (h) (j) (k), 17.6, 17.8, 17.15 (b), 17.17], 17.9 (a), 17.18.
Section 18: [Think: 18.1 (a) (c), 18.2], [Examples: 18.3 (a) (d), 18.4], 18.5 (algorithm for computing the square root of x>0), 18.6 (geometric series, decimal expansion).

Homework 9 (due Nov 8)

Section 16: [Think: 19.1,19.2, 19.7, 19.10, 19.16], 19.3(b)(c)(d), 19.4(a)(b), 19.5(d), 19.8, 19.12, 19.13, 19.14.

Homework 10 (due Nov 27)

Sec 20: [think 20.1, 20.2, 20.11, 20.12], 20.3(b)(f)(h), 20.6(b), 20.7, 20.14, 20.15, 20.18. Sec 21: [think 21.1, 21.2, 21.7, 21.10], 21.3, 21.4, 21.5, 21.6, 21.11, 21.16.

Homework 11 (due Dec 4)

Sec 22: [think 22.1, 22.2, 22.3], 22.4, 22.8, 22.11, 22.13, 22.16. Sec 23: [think 23.1, 23.2] 23.3 (use the theorems in the section to justify your answer, not the definition), 23.4(a)(b), 23.5, 23.9, 23.10, 23.12. Sec 25: [think 25.1, 25.2] 25.3(a)(c)(d), 25.5, 25.6, 25.9.