# Lecture Schedule / Homework Assignments

**Mon 12/8: Final Exam**1-3pm in Kariotis 209.**Wed 12/3:**Review for final exam.**Homework 7 due in class.**Practice problems for the final exam, with solutions.

**Mon 12/1:**Ch 5.6: The Arzela-Ascoli Theorem.**Wed 11/26: Thanksgiving break: no class.****Mon 11/24:**Ch 5.5: The space of continuous functions.**Homework 6 due in class.**Office hours will be from 11:30-1pm today instead of the usual 11-12:30, as I have to cover lectures for another professor this morning.

Homework 7 will be due on

**Wednesday 12/3**.**Wed 11/19:**Ch 5.1: Pointwise vs. uniform convergence of functions.**Mon 11/17:**Ch 4.8: Integration.Homework 6 will be due on

**Monday 11/24**.**Errata:**In class I tried to define the Cantor function but did not do so correctly. Wikipedia has a decent article on the actual Cantor function. I also asserted incorrectly that the Cantor function was not continuous; in fact it*is*continuous, though this is not immediately obvious.**Wed 11/12:**Ch 4.7: Differentiation continued. Lipschitz continuity.**Mon 11/10:**Ch 4.6: Uniform continuity, start of Ch 4.7: Differentiation.**Homework 5 due in class.**Here is the proof of the chain rule that I gave in class.

**Mon 11/3 and Wed 11/5:**Ch 3.4, 3.5, 4.2, 4.5: Connected and path-connected sets, images of connected sets and the intermediate value theorem.I am out of towm, so Prof. Beasley will give the lectures this week, and there will be no office hours. You can email me with questions, or attend Prof. Suciu's office hours (who teaches the other section of analysis).

Homework 5 will be due on

**Monday 11/10**.**Wed 10/29:**Ch 3.2, 3.3, 4.2: Compact sets in R^n, nested set property, compact sets and continuous functions.**Homework 4 due in class.****Mon 10/27:**Ch 4.3, 3.1: Operations on continuous functions, Compact sets and the Bolzano-Weirstrauss theorem.**Wed 10/22:**Ch 4.1: Continuous functions.Homework 4 will be due on

**Wednesday 10/29**.Here are Solutions to the midterm.

**Mon 10/20: Midterm exam.****Wed 10/15:**Review for midterm.~~If time, some of Ch 2.9: Series in R^n.~~Midterm will cover Ch. 1.1-1.7, except for liminf/limsup (second half of 1.5), and Ch. 2.1-2.8

In the exam review, I attempted the problem: "If A and B are closed sets in R^n, is A + B closed? Give a proof or a counterexample." I had thought that A + B was closed, but failed after several different attempts to prove it.

For those of you interested in the right answer: A + B need not be closed! While I also failed to come up with a counterexample, I eventually found one online. Let A be the natural numbers and B be the set of (-n + 1/n) for all natural numbers n. A and B are closed in R, but 0 is an accumulation point of A + B which is not in A + B. If you add the hypothesis that one of A or B is also

*bounded*, then A+B will be closed. We'll be able to prove this once we've talked about compact sets.**Mon 10/13: Columbus Day (no classes).****Wed 10/8:**Ch 2.8: Completeness.~~Ch 2.9: Series in R^n.~~**Homework 3 due in class.**The

**midterm exam**will be in class on Monday October 20 (one week from next Monday).**Mon 10/6:**Ch 2.6-2.8: Boundary of a set, sequences and completeness in metric spaces and R^n.**Wed 10/1:**Ch 2.2-2.5: Interior of a set, closed sets, accumulations points, closure of a set.Homework 3 will be due on

**Wednesday 10/8**.**Mon 9/29:**Ch 2.1: Open sets,~~(if time) Ch 2.2: Interior of a set.~~**Wed 9/24:**Ch 1.7: Norms, inner products and metrics.**Homework 2 due in class,**or by 5pm Thu 9/25 by electronic submission over email.**Mon 9/22:**Ch 1.5: Cluster points,~~liminf and limsup.~~, Ch 1.6: Euclidean space.In a last minute change, I decided to skip the material on limsup and liminf (apart from their definition), and instead start discussing Euclidean space, as a vector space with an inner product.

**Wed 9/17:**Ch 1.4: Cauchy sequences.Homework 2 will be due on

**Wednesday 9/24**.Here are Solutions to Homework 1.

Here is a nice set of notes on the construction of R from Q by Cauchy sequences, by Professor Todd Kemp of UCSD.

The final exam dates have been posted. Our final exam will be on Monday December 8, from 1-3pm. Please check as soon as possible to make sure you don't have any conflicts.

**Mon 9/15:**Ch 1.3: Least upper bounds.**Homework 1 due in class.**Please remember to cite your collaborators on your homework.Here is a note containing the proof given in class that nth roots of positive real numbers exist.

**Wed 9/10:**Ch 1.2: Limits and completeness, continued.As announced in class, my office hours going forward will be changed to Mondays 11:00-12:30 and Wednesdays 1:30-2:50, or of course by appointment.

**Mon 9/8:**Ch 1.2: Limits and completeness.Homework 1 will be due on

**Monday 9/15**.**Wed 9/3:**Ch 1.1: Ordered fields.Here are some notes about proof by induction and contradiction, and using these to prove the well-ordering property of the natural numbers.