Date
|
Topic
|
Resources
|
June 22
|
Introduction
|
-
Notes.
-
Rudin, Chapter 1, everything up through the end of the section on
“Fields.”
-
Ross, Chapter 1, everything before 4.4.
|
June 23
|
Real numbers
|
-
Notes.
-
Rudin, Chapter 1, “The Real Field” and “The Extended Real Number
System.”
-
Ross, Chapter 1, from 4.4 up through the end of section 5.
-
For examples of nonarchimedean ordered fields, check
this
out, or
this
for a list of further links to examples.
|
June 24
|
Metric spaces
|
|
June 25
|
Open subsets (continued), Closed subsets
|
-
Notes.
-
Rudin, Chapter 2, “Metric spaces.”
-
Ross, parts of section 13.
|
June 29
|
Closed subsets (continued), Connected spaces
|
-
Notes.
-
Rudin, Chapter 2, “Connected sets.”
-
Ross, section 22, up through example 1.
|
June 30
|
Compact spaces
|
-
Notes.
-
Rudin, Chapter 2, “Compact sets.”
-
Ross, parts of section 13.
|
July 1
|
Compact spaces (continued)
|
|
July 2
|
Compact spaces (continued), Cantor set
|
-
Notes.
-
Rudin, section 2.44.
-
Ross, section 13, example 5.
|
July 6
|
Sequences
|
-
Notes.
-
Rudin, chapter 3, “Convergent sequences” up through the proof of theorem
3.2, “Subsequences” except theorem 3.6, and Cauchy sequences up through
theorem 3.11(a).
-
Ross, parts of sections 7, 9 and 10 (but with arbitrary metric spaces
instead of just real numbers), and also most of the parts of section 13
we haven’t discussed already.
|
July 7
|
Compactness and sequences
|
-
Notes.
-
Rudin, chapter 3, theorems 3.6 and 3.11.
-
Ross, theorems 10.11, 13.4, 13.5
|
July 8
|
Compactness and sequences (continued), Sequences in
R
|
-
Notes.
-
Rudin, chapter 3, statement of theorem 3.3, definition 3.13 and theorem
3.14, and theorem 3.20.
-
Ross, section 9 and 10, except the bit about limsup and liminf in
section 10.
|
July 9
|
Limit superior and limit inferior
|
-
Notes.
-
Rudin, chapter 3, “Upper and lower limits.”
-
Ross, the bit about limsup and liminf in section 10, and then section
12.
|
July 13
|
Series
|
-
Notes.
-
Rudin, chapter 3, “Series,” “Series of nonnegative terms,” “The root and
ratio tests,” “Power series,” “Absolute convergence,” and
“Rearrangements.”
-
Ross, sections 14, 15 and 23.
|
July 14
|
Series (continued)
|
|
July 15
|
Series (continued), Continuity
|
-
Notes.
-
Rudin, chapter 4, “Limits of functions,” “Continuous functions,” and
“Continuity and connectedness.”
-
Ross, sections 17, 20 and 21.
|
July 16
|
Continuity (continued)
|
|
July 20
|
Continuity (continued), Uniform continuity
|
-
Notes.
-
Rudin, 4.18.
-
Ross, section 19.
|
July 21
|
Continuity and compactness
|
-
Notes.
-
Rudin, chapter 4, “Continuity and compactness.”
-
Ross, sections 18 and 19.
|
July 22
|
Space of continuous functions
|
-
Notes.
-
Rudin, chapter 7, up through “Uniform convergence and continuity.
-
Ross, sections 24 and 25.
|
July 23
|
Differentiation
|
-
Notes.
-
Rudin, chapter 5, up through “Mean value theorems,” and chapter 7,
“Uniform convergence and differentiation.”
-
Ross, sections 28 and 29.
|
July 27
|
Differentiation (continued)
|
|
July 28
|
Differentiation (continued), Applications of differentiation
|
-
Notes.
-
Rudin, chapter 5, “L’hospital’s rule” and “Taylor’s theorem.”
-
Ross, sections 30 and 31.
|
July 29
|
Applications of differentiation (continued)
|
|
July 30
|
Integration
|
-
Notes.
-
Rudin, chapter 6, up through theorem 6.13 (except assuming that alpha(x)
= x), and theorem 7.16.
-
Ross, sections 32 and 33.
|
August 3
|
Integration (contined)
|
|
August 4
|
More on integration
|
-
Notes.
-
Rudin, chapter 6, ``Integration and differentiation.’’
-
Ross, sections 26 and 34.
|