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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.