# Math 104

## Introduction to Analysis

Lectures MWF 13:10-14:00, 71 Evans Hall

Office hours WF 14:00-15:00, 821 Evans Hall

#### Topics covered

• 08/29: Basic notation and terminology. Integers and rationals.
• 08/31: Ordered sets. Upper and lower bounds. Suprema and infima.
• 09/02: Fields. Ordered fields.
• 09/07: The real number system.
• 09/09: The complex field.
• 09/12: Euclidean spaces. Finite and infinite sets.
• 09/14: Basic set operations. Countable and uncountable sets.
• 09/16: Metric spaces. Open and closed sets.
• 09/19: Limit points. Interior and closure.
• 09/21: Relatively open and closed sets. Compactness.
• 09/23: More on compactness.
• 09/26: The Heine-Borel theorem and consequences.
• 09/28: Perfect sets. Connected sets.
• 09/30: Convergent sequences.
• 10/03: Subsequences.
• 10/05: Cauchy sequences.
• 10/07: Upper and lower limits. Series.
• 10/10: Nonnegative series. The dyadic trick. The number e.
• 10/12: Root and ratio tests. Power series.
• 10/14: Summation by parts. Absolute convergence.
• 10/17: Cauchy product of series. Rearrangements.
• 10/19: Midterm on Chapters 1-3 of the book.
• 10/21: Discussion of the midterm.
• 10/24: Continuous maps of metric spaces.
• 10/26: Continuity and compactness.
• 10/31: Continuity and connectedness. Discontinuities.
• 11/02: Monotonic functions. Infinite limits and limits at infinity.
• 11/04: The derivative of a real function.
• 11/07: Mean value theorems. Continuity of derivatives.
• 11/09: L'Hospital's rule. Taylor's formula.
• 11/14: Differentiation of vector-valued functions.
• 11/16: Riemann-Stieltjes integral.
• 11/18: Riemann sums, refinements, and the integrability criterion.
• 11/21: Integrating continuous and monotone functions.
• 11/23: Properties of the integral. Integrating against step functions.
• 11/28: The fundamental theorem of Calculus (3 versions).
• 11/30: Integration of vector-valued functions. Rectifiable curves.
• 12/02: Interchanging limit processes. Negative examples. Uniform convergence.
• 12/05: Uniform convergence and continuity. Uniform convergence and integration.
• 12/07: Uniform convergence and differentiation. A nowhere differentiable continuous function.
• 12/09: Using MATLAB for real analysis.

### Resources

• Books.
• Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill.
• Charles Pugh, Real Mathematical Analysis, Springer-Verlag (2002).

The first book will be the main textbook for the course.

• Darren Rhea's Math 104 website.

The instructor welcomes cooperation among students and the use of books. However, handing in homework that makes use of other people's work (be it from a fellow student, a book or paper, or whatever) without explicit acknowledgement is considered academic misconduct.

### Assignments

All homework problems are from the 3rd edition of Rudin.

• Homework assignment #1, due Sep 7th: Exercises 1, 2, 4, 5, 8 (p.21).
• Homework assignment #2, due Sep 14th: Exercises 9, 11, 13, 14, 17 (p.22).
• Homework assignment #3, due Sep 21st: Exercises 2, 4, 5, 8 (p.43).
• Homework assignment #4, due Sep 28th: Exercises 11, 12, 14, 17 (p.44).
• Homework assignment #5, due Oct 5th: Exercises 20 (p.44), 1, 2, 3, 4 (p.78).
• Homework assignment #6, due Oct 12th: Exercises 5, 6, 7, 8 (p.78).
• Homework assignment #7, due Oct 19th: Exercises 9, 11, 13 (p.78).
• Homework assignment #8, due Nov 2nd: Exercises 1, 2, 3, 4 (p.98).
• Homework assignment #9, due Nov 9th: Exercises 8, 14, 16, 18 (p.98).
• Homework assignment #10, due Nov 16th: Exercises 1, 4, 6, 12, 17 (p.114).
• Homework assignment #11, due Nov 23rd: Exercises 15 (p.115), 1, 2, 4 (p.138).
• Homework assignment #12, due Nov 30th: Exercises 5, 11, 15 (p.138).
• Homework assignment #13, due Dec 9th: Exercises 1, 2, 4 (p.165).

### Solutions

• Solutions to homework assignment #1, in PS and in PDF.
• Solutions to homework assignment #2, in PS and in PDF.
• Solutions to homework assignment #3, in PS and in PDF.
• Solutions to homework assignment #4, in PS and in PDF.
• Solutions to homework assignment #5, in PS and in PDF.
• Solutions to homework assignment #6, in PS and in PDF.
• Solutions to homework assignment #7, in PS and in PDF.
• Solutions to homework assignment #8, in PS and in PDF.
• Solutions to homework assignment #9, in PS and in PDF.
• Solutions to homework assignment #10, in PS and in PDF.
• Solutions to homework assignment #11, in PS and in PDF.
• Solutions to homework assignment #12, in PS and in PDF.
• Solutions to homework assignment #13, in PS and in PDF.

• Assignment #10: Ex 1: Almost everyone had some trouble with the absolute value sign. Most want to say that the absolute value of f'(y) is the limit as x goes to y of the absolute value of (f(x)-f(y))/(x-y), or something like that. I only gave full credit if they showed that this last limit equals 0 (via squeeze), and then conclude that the limit of (f(x)-f(y))/(x-y) is 0, or something equivalent. Ex 4: Some people tried using integrals, which I think is equivalent to using derivatives on the antiderivative, the more common way. They were unable to rigorize the argument using integrals, though. Ex 6: This is mostly all or nothing. Ex 12: Most people didn't realize they had to explicitly check that f'(0) and f''(0) exist (e.g. by noting that the one-sided limits agree). Some people decided to ignore the absolute value when differentiating and ended up with f'(x) = 3|x|^2 (incorrect), f''(x) = 6|x| (correct but faulty logic to get here). Ex 17: Almost everyone proved the hint correctly, but many did not prove why it implies either f^(3)(s) or f^(3)(t) must be greater than or equal to 3.

• Assignment #11: 15. I thought it was unclear what the problem actually wanted you to show, so I gave full credit for proving the inequality on page 115 without regard to equality. Several students tried to finish the proof by presenting a quadratic in h that is always nonpositive for h > 0, and concluding that the discriminant must be nonpositive. This isn't quite correct because we don't know what happens when h < 0. Now what happens is that the quadratic and linear terms are of opposite signs, so the discriminant argument can be fixed. I feel bad for marking off for a good idea that needs fixing up. The proofs that involve substitute a nice value for h are more successful. 1. Some students thought this is just a Riemann integral and failed to account for the alpha part. 2. Many arguments asserted that the integral of f from a to b is greater than or equal to the integral of f on a smaller interval. Basically no one bothered to justify this step (which involves breaking up an integral and then doing a comparison with 0). 4. Many students derived that U(P,f,alpha) - L(P,f,alpha) = b-a for any partition P, but then did not really finish the argument. There seems to be some confusion on applying Theorem 6.6: some students think that it implies that U = L for some partition P.

• Assignment #12: In exercise 15, almost no one realized that after using Schwartz, one still has to prove that strict equality cannot occur. First, xf(x) and f'(x) must be proportional. Then one can solve the differential equation (which produces an exponential that's nowhere or everywhere zero). Or there's an interesting approach one student tried (rigorized by me), involving the following three facts: 1. If u,v are distinct zeroes of f with u < v, then there exists another zero t such that u < t < v; apply mean value theorem, 0 = f(v)-f(u) = (v-u)f'(t) = (v-u)ctf(t), so f(t) = 0. 2. The kernel of f is closed (by continuity). 3. The kernel of f contains a and b. (These facts imply that the kernel of f is the whole interval [a,b].)

• Assignment #13: 1. Suppose the functions f_n converge to f uniformly. One can show that f is bounded and thus the f_n's after a certain point are also uniformly bounded. Or one can show the latter directly by using Cauchy's criterion (Theorem 7.8). Quite a few people used stuff like |f_n - f| -- I think they meant to use the function norm but failed to use the double bar notation. I let this slide because the damage is coming later. 2. This problem basically had one canonical solution. Not many people realized that exercise 1 is needed to get a uniform bound on f_n and g_n in the second part. 3. This problem is quite hard. Details will be given in the solution.