Math 131: Introduction to Analysis


In this course we will begin a study of real analysis, including basic properties if the real numbers, sequences and series, various notions of convergence, differentiation and integration.


Dagan Karp
Office hours: Tue 4:00-5:00 pm, and Thu 3:00-4:00 pm by appointment, and open door.
Office Location: Shanahan 3414

Tutors and Graders

Emily Fischer, Alexander Gruver and Ben Lowenstein are tutoring and grading for this course. You should talk to them frequently and take advantage of their expertise! Tutoring hours are:

Sunday: 7-9 pm
Tuesday: 7-9 pm
Tutoring will be held in Sprague 1.


The official text for this course is Principles of Mathematical Analysis, by Walter Rudin.


There will be two midterms, one final exam and weekly homework. Each will count one fifth of your grade. The remaining fifth is the maximum of these four items.


Written homework will be due in class each Wednesday, and is posted below. Late homework may not be accepted. Each student's lowest homework score will be dropped. Please consult the HMC mathematics homework format guidelines for helpful tips on homework submission and formatting.


Students have the opportunity to turn in a rewritten copy of the first four homeworks for a revised grade. Details can be found on the Math 131 Rewrite Policy.


There will be three exams. The first exam will be in-class on Monday, February 24. The second exam will be take-home, out Wed March 26 and due Mon March 31. The third exam will be either take-home or in-class, but will be due at 2pm on Wednesday, May 14.


Students should feel free to use LaTeX for homework, but this isn't required.


Students who need disability-related accommodations are encouraged to discuss this with the instructor as soon as possible.

Honor Code

Though cooperation on homework assignments is encouraged, students are expected to write up their own solutions individually. That is, no copying. Comprehension is the goal, so even with cooperation, you should understand solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. Tests are to be done individually. You are encouraged to discuss your paper with other students. The HMC Honor Code applies in all matters of conduct concerning this course.

LaTeX Lecture Notes

These lecture notes are generously provided by Julius Elinson. Beware they may contain typo's and mistakes coming directly from the blackboard!

Lecture 2 PDF, TEX
Lecture 3 PDF, TEX
Lecture 4 PDF, TEX
Lecture 5 PDF, TEX
Lecture 6 PDF, TEX
Lecture 7 PDF, TEX
Lecture 8 PDF, TEX
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Lecture 22 PDF, TEX
Lecture 23 PDF, TEX

iPad Lecture Notes

These lecture notes are generously provided by Hannah Rose. As always, beware of typos!

Lecture 1: Construction of Q, introduction to Fields, example of an irrational number.
Lecture 2: Suprema, Infima and Dedekind Cuts
Lecture 3: Extensions of R, Definition of C, and Introduction to Countable Sets
Lecture 4: Countable and Uncountable Sets
Lecture 5: Introduction to Metric Spaces, Neighborhoods, Interior Points, Limit Points
Lecture 6: Open and Closed Sets, Closure and Interior, Definition of a Topology
Lecture 7: Subspace Topology, Compact Sets
Lecture 8: Heine-Borel and Balzano-Weierstrass
Lecture 9: Connected Sets, Introduction to Sequences
Lecture 10:Basic properties of sequences, Cauchy sequences
Lecture 11: Subsequences, upper and lower limits, complete metric spaces, the completion of a metric space
Lecture 12: Properties of upper and lower limits, special sequences, definition of infinite series as a sequence of partial sums
Lecture 13: Infinite Series, Cauchy Criterion for Series, Divergence Test, Comparison Test, Geometric Series, the number e
Lecture 14: p-serier, Ratio Test, Root Test, Rearrangements, Alternating Series
Lecture 15: Limits and Continuity
Lecture 16: Continuity Continued, Topological Characterization, Continuous Images and Compact and Connected Sets
Lecture 17: Continuous Images of Connected Sets, Continuous Inverses of Bijections on Compact Sets, Intermediate Value Theorem, Uniform Continuity
Lecture 18: Lebesgue Number Lemma, Discontinuities of the First and Second Kind, Pointwise Limits of Sequences and Series of Functions
Lecture 19: Uniform Convergence of a Sequence of Functions, Introduction to Differentiability
Lecture 20: Derivatives Continued, Mean Value Theorem

YouTube Lectures

Francis Su delivered these lectures in 2010. My lectures will be very different, but you may find these valuable for review, or, better yet, watch them before the class lecture, and then during class you can ask questions!

Lecture 1: Constructing the rational numbers
Lecture 2: Properties of Q
Lecture 3: Construction of R
Lecture 4: The Least Upper Bound Property
Lecture 5: Complex Numbers
Lecture 6: The Principle of Induction
Lecture 7: Countable and Uncountable Sets
Lecture 8: Cantor Diagonalization, Metric Spaces
Lecture 9: Limit Points
Lecture 10: Relationship b/t open and closed sets
Lecture 11: Compact Sets
Lecture 12: Relationship b/t compact, closed sets
Lecture 13: Compactness, Heine-Borel Theorem
Lecture 14: Connected Sets, Cantor Sets
Lecture 15: Convergence of Sequences
Lecture 16: Subsequences, Cauchy Sequences
Lecture 17: Complete Spaces
Lecture 18: Series
Lecture 19: Series Convergence Tests
Lecture 20: Functions - Limits and Continuity
Lecture 21: Continuous Functions
Lecture 22: Uniform Continuity
Lecture 23: Discontinuous Functions
Lecture 24: The Derivative, Mean Value Theorem
Lecture 25: Taylor's Theorem
Lecture 26: Ordinal Numbers, Transfinite Induction

Homework Assignments

Homeworks, due Thursdays in class
All HW's refer Rudin's Principles of Mathematical Analysis.

  • HW #0. Due Monday Jan 27. Read this handout on good mathematical writing and turn in brief answers to these questions.

    • Directly from the handout reading:
      • 1. What is a good rule of thumb for what you should assume of your audience as you write your homework sets?
      • 2. Is blackboard writing formal or informal writing?
      • 3. Do you see why the proof by contradiction on page 3 is not really a proof by contradiction?
      • 4. Name 3 things a lazy writer would do that a good writer wouldn't.
      • 5. What's the difference in meaning between these three phrases?
          "Let A=12."
          "So A=12."
    • Now examine Section 1.1 of Rudin, showing that there is no rational p that satisfied p2=2.
      • 6. There are many places in his proof where he could have used symbols to express his ideas, but he does not. (e.g., "Let A be the set of all positive rationals p such that...") Why do you think he chooses not to use symbols?
      • 7. What would you change about his presentation if you were writing for a high school audience? Give a specific example.

    Keep in mind the handout and the homework format as you write up your answers.

  • HW #1. Due Jan 29. Read Chapter 1 in its entirety. Do not worry about understanding everything, just read for the big ideas. Turn in the following problems:


      Problem A. In no more than four sentences, describe the main themes and concepts of Chapter 1. The first sentence or two should be at a level that your parent could understand even if they never went to college. The other sentences should be understandable by any college student.
      Problem B. Recall that in class, we defined a rational number m/n to be an equivalence class of pairs (m,n) under an equivalence relation. We also defined addition of rational numbers in terms of representatives: a/b + c/d = [ad+bc]/[bd]. Show that the addition of rational numbers is well-defined.
      Problem C. Define a multiplication of rational numbers, and show this multiplication is well-defined.


      Do also Chapter 1 ( 1, 2, 3a ).

    Please hand in PARTS ALPHA and BETA in two separately stapled parts. (ALPHA and BETA parts will be graded separately.) Each part should have your name and follow this homework format as well as the guidelines for good mathematical writing.

    In all homework, remember to concentrate on good writing.

p.s. Be sure to subscribe to math-131-l (using if you are not already getting e-mails from the list.

  • HW #2. Due Feb 5. A problem marked "R" means read, but do not do the problem.


      Chapter 1 ( R3bcd, R4, 5, R7, 8, 9 ).


      Chapter 2 ( 2, R3, 4 ) and
      Problem S. For a real number a and non-empty subset of reals B, define: a + B = { a + b : b is in B }. Show that if B is bounded above, then sup( a + B ) = a + sup B.

p.s. Be sure to subscribe to math-131-l (using if you are not already getting e-mails from the list.

  • HW #3. Due Feb 12.

    Read Chapter 2.


      Do Chapter 2 ( 5, R6, R7, 8, 11[exclude d2] ).


      Do Chapter 2 ( 9ab, 9cd, 9ef, R10 )

  • HW #1 Rewrite Due. See the rewrite policy for details.

  • HW #4. Due Feb 19. A problem marked "R" means read, but do not do the problem. There are no re-writes on assignments past HW 4; this is the last one.
    See the rewrite policy for details.


      Do Chapter 2 ( 7, 12, R14, 22, R23 ).


      Do Chapter 2 ( 16[closed,bounded], 16[not compact, open?], R23, 25 ). Problem 25 requires you to read Problem 23.

  • HW #2 Rewrite Due

  • EXAM 1. In class, Monday Feb 24.

  • HW #5. Due Mar 5. A problem marked "R" means read, but do not do the problem.
    HW 5 may not be re-written, nor any further assignments.
    See the rewrite policy for details.


      Do Chapter 2 ( R15, 17[countable?, dense?], 17[compact?, perfect?], 18 ).


      Do Chapter 2 ( 19, 20, 24, R26 ).

  • HW #3 Rewrite Due

  • HW #6. Due Mar 12. A problem marked "R" means read, but do not do the problem.


      Do Chapter 3 ( 1, 3, R16, 20 ) Hint on 3.3: can you show the sequence is increasing? Induction may be of help here.


      Do Chapter 3 ( 2, 5, 23 )

  • HW #4 Rewrite Due

  • HW #7. Due March 26. A problem marked "R" means read, but do not do the problem.


      There is no part alpha this week. Enjoy Spring Break.


      Do Chapter 3 ( 24ab, 24cde, 25 )

  • EXAM 2. Due Mon March 31.

    Review your notes in preparation for the midterm.

  • HW #8. Due Apr 9. A problem marked "R" means read, but do not do the problem.


      Chapter 4 ( 1, 2, 3 )


      Do Chapter 4 ( 4, 5, R6, R7, 8 )
    On the reading problem 4.6, you may assume that E is a subset of real numbers and f is a real-valued function, and the distance in R2 (where the graph lives) is the usual Euclidean metric.

  • HW #9. Due Apr 16. A problem marked "R" means read, but do not do the problem.


      Do Chapter 4 ( 10, 11* ). *You do NOT have do the 2nd part of Problem 4.11, where it says "Use this result to give..."


      Do Chapter 4 ( 12, 14, R16, 18, R19 ).

  • HW #10. Apr 23.


      Do Problem E: Use the mean value theorem to show that e^x is greater than or equal to (1+x) for all x in R. (You may assume knowledge of the derivative of e^x.)

      and Chapter 5 ( 1, 4 ).


      Chapter 5 ( 13ab, R13cdefg, R25abd, 25c ) and Chapter 7 ( R2, R3, 7 ). In Problem 5.13, there is a typo (in some editions of the book): the xa should say |x|a.

  • HW #11. Due Apr 30.


      Do Chapter 6 ( 1, 2, 4 ).


      Chapter 6 (5, 8)

    Exam 3.
    Non-Seniors: Due May 14 at 2pm.
    Seniors: Due May 8 at 5pm.

Possible upcoming homeworks

Below this line, all homeworks are TENTATIVE. This means they are likely to be assigned, but there is no guarantee that they will until you see them moved to the box ABOVE. I am putting them here in case you want to work ahead!