Asst #

Topic

Reading

Problems

Ch 1

The Real Numbers

1

Introduction and Preliminaries

pp. v-x; 1-13

p. 11 # 1.2.3ab, 1.2.5, 1.2.10, 1.2.11, 1.2.12

2

Completeness Axiom and Its Consequences

pp. 13 - 26

p. 17 #1.3.1, 1.3.2, 1.3.3, 1.3.6, 1.3.9,

p.27 #1.4.2, 1.4.4,1.4.5, 4.6a, 1.4.8, 1.4.9, 1.4.12

3

Cantor's Theorem and Power Sets

pp. 26-34

p.30 #1.5.1, 1.5.2, 1.5.5, 1.5.8, 1.5.9ab

Ch 2

Sequences and Series

4

Infinite Sequences and Limits

pp. 35-43

p.43 #2.2.1ab, 2.2.2, 2.2.5,  2.2.7a,  2.2.7b

5

Limit Theorems

pp. 44-51

p.49 #2.3.1, 2.3.2, 2.3.3, 2.3.4, 2.3.8

6

Infinite Series

pp. 51-55

p.54 #2.4.2, 2.4.4, 2.4.5a, 2.4.6abc,2.4.6d

7

Bolzano-Weierstrass;

Cauchy's Criterion

pp. 55-62

p.57 #2.5.1, 2.5.2, 2.5.3, 2.5.4, 2.6.1

8

Properties of Infinite Series

pp. 62-69

p. 67 #2.7.1b, 2.7.2a,

2.7.3, 2.7.4, 2.7.9, 2.7.11

Ch 3

Basic Topology of R

9

Topology of the Reals

pp. 75-79

p. 82 #3.2.1, 3.2.2, 3.2.4, 3.2.5

10

Limit Points and Closed Sets;

Compact Sets

pp. 79-86

p. 83 #3.2.6, 3.2.8, 3.2.9, 3.2.10ab; 3.2.13

 p. 87 #3.3.1 3.3.2, 3.3.3, 3.3.4

Ch 4

Functional Limits and Continuity

11

Functional Limits and Continuity

pp. 99-105

p. 108 #4.2.1ac, 4.2.2, 4.2.3ab, 4.2.4

12

Sequential Criterion for Functional Limits

Algebra of Continuous Functions

pp. 105-114

p. 108 #4.2.5ab, 4.2.5c, 4.2.6. 4.2.7abc, 4.2.9;

p. 113 # 4.3.1, 4.3.2a, 4.3.2b, 4.3.4a, 4.3.4b, 4.3.5, 4.3.7, 4.3.8

13

Compactness and Uniform Continuity

pp. 114-119

p. 113 #4.3.10 

p. 119 # 4.4.2, 4.4.3, 4.4.6, 4.4.7, 4.4.8a

14

Intermediate Value Theorem; Discontinuities

pp. 120-128

p. 124 # 4.5.2, 4.5.4, 4.5.5 

p.125 # 4.6.1ab, 4.6.2

Ch 5

The Derivative

15

Definitions and Examples

pp. 129-132

p. 126 #4.6.5

p. 136 #5.2.3, 5.2.4

16

Chain Rule; DarbouxÕs Theorem;

Mean Value Theorems

pp. 133-143

p. 136 #5.2.5, 5.2.6, 5.2.7, 5.2.8abc;

p. 143 #5.3.3abc, 5.3.4, 5.3.5, 5.3.7a, 5.3.7b

17

Continuous Nowhere Differentiable

pp. 144-149

p. 144 # 5.3.11

p. 147 #5.4.5a

Ch 6

Sequences and Series of Functions

18

Introduction: Branching Processes;

Pointwise and Uniform Convergence

pp. 151-160

p. 160 #6.2.1a, 6.2.2a, 6.2.3a; 6.2.1bcd, 6.2.2b, 6.2.3b, 6.2.6

19

Uniform Convergence and Differentiation; Series

pp. 164-169

p. 166 #6.3.1ab, 6.3.2, 6.3.5, p. 168 #6.4.1, 6.4.2

20

Power Series and Taylor's Theory

pp. 169-173

pp. 176-180

p. 174 #6.5.1, 6.5.2, 6.5.5

p. 177 #6.6.1, 6.6.2, 6.6.3

Ch 7

The Riemann Integral

 21

Definitions of Integral

 pp. 183-188

p. 190 #7.2.1, 7.2.2, 7.2.3

22

Integrability;

Integrating with Discontinuities

 pp. 188-193

p. 191 #7.2.4abc, 7.2.5, 7.2.6;

p. 193 #7.3.1abc, 7.3.2, 7.3.4, 7.3.5

23

Uniform Convergence; Fundamental Theorem of Calculus

pp. 195-198

pp. 199-201

p. 191 #7.2.5  p. 199  #7.4.4 pp. 201 #7.5.1

24

Lebesgue's Criterion

pp. 203-207

p. 204 #7.6.1, 7.6.2, 7.3.6c (note!) and 7.6.4

Take-home final exam problems

Due

Dec 16 at 10:00 pm