# Overview

• Class is Monday through Thursday, 10-12 in 107 GPB.
• Office hours are in 1043 Evans on Mondays and Wednesdays, 12:30-1:30, and Tuesdays, 9-10.
• Syllabus

# Announcements

• All revisions on problem sets 1 through 6 must be submitted by Thursday, 8/6.
• On problem set 2, for problem 8, the set C should exclude the empty set.
• My office hours on Wednesday, 7/8, will end promptly at 1:15.
• Problem set 1 has a very minor correction: for problem 3, one should assume that S and T are nonempty.
• Starting Tuesday 6/30, my Tuesday office hours are moved to 9-10 instead of 12:30-1:30. This change is reflected above.
• My office hours on Tuesday 6/23 are dedicated for reviewing the following concepts: sets, subsets, unions, intersections, cartesian products, relations, functions, injectivity, surjectivity, bijectivity, having the same cardinality, countability, and uncountability. Come if you are unfamiliar with any of these concepts, or would just like to review some of these concepts with me! (Note that you can also read a very terse introduction to many of these topics in the first section “Finite, Countable and Uncountable Sets” in chapter 2 of Rudin.)

# Calendar

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)
• See above.
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)
• See above.
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)
• See above.
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)
• See above.
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)
• See above.
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)
• See above.
August 4 More on integration
• Notes.
• Rudin, chapter 6, Integration and differentiation.’’
• Ross, sections 26 and 34.