MAT 127A: Real Analysis (Fall 2019)
Course Materials

  • Some writing tips I wrote for MAT 108.
  • Sample exams from an earlier version of this course, which I taught 5 years ago: Midterm 1, Midterm 2, Final.
  • Discussion Problems 1.
  • Discussion Problems 2.
  • Discussion Problems 3.
  • Discussion Problems 4. Solutions.
  • Discussion Problems 5.
  • Midterm 1 will be on Fri., Oct. 25, 2019, in class. It covers Sections 1.3-1.4, 2.2-2.5 from the book, and first four homework assignments. It also covers infinite limits, limsup and liminf (see Prof. Hunter's lecture notes on sequences for a discussion on liminf and limsup). Topics: sup and inf, max and min, completeness axiom and consequences, definition of limit of a sequence (and proofs using that definition), algebraic and order limit theorems, monotone sequences and their convergence, subsequences, Bolzano-Weierstrass theorem, infinite limits, limsup, liminf. You will need to understand examples covered in the lecture and in the discussion, as well as on the homework assignments.
    For exam practice, consult Discussion Problems 5. These problems will be discussed during a Discussion Session.

    Solutions to Midterm 1.

  • Discussion Problems 6.
  • Discussion Problems 7.
  • Discussion Problems 8.
  • Discussion Problems 9.
  • Midterm 2 will be on Fri., Nov. 22, 2019, in class. It covers Sections 2.7, 3.2, 3.3, 3.4 (connected sets only) from the book, and homework assignments 5-8. You also need to know the material covered on the first midterm, although there will be no problems that specifically test that. This exam will not feature any questions specifically testing the material from Section 2.6 (Cauchy criterion). Topics: convergence of infinite series (definition, absolute convergence, relative convergence, algebraic and order theorems, various tests for convergence), alternating series, rearrangements, geometric series, p-series, open sets, closed sets, accumulation (limit) points, isolated points, interior points, boundary points, closure of a set, compact sets (definition, Heine-Borel, open covers), connected sets.
    For exam practice, consult Discussion Problems 9. These problems will be discussed during a Discussion Session.

    Solutions to Midterm 2.

  • Discussion Problems 10.
  • Practice Final Exam.
  • Finals Week Info