- Not quite solutions, but very strong hints added for the 2007 exam below.
- 12 December, 2010; Question from Gus Downs:- I believe this discussion has come up on the last two tests as well, but I can't remember what was decided. Are we allowed to cite theorems from Rudin freely in our test answers, or are we required to reprove them if we do? Thank you for your time.

My answer:- Absolutely you are allowed to cite theorems (no numbers or anything like that required) as long as you can state them clearly and correctly. So, not just `by a theorem in Rudin we can do X' but `A theorem in Rudin states that every continuous function on a compact metric space is uniformly continuous, so X' -- the only exception would be if you are actually asked to prove something which is in fact a theorem in Rudin in which case you have to produce something like the proof in the book or one I would give on the blackboard (but better maybe ..). - Since I haven't found solutions to the 2007 exam (and I may not have the time to write them up) I added another old exam with solutions, 10 December, 2010. Note that the last question in the 2002 exam is not really relevant since I have not talked about contractions this time.
- I put up two old final exams below which should give you a good idea of what to expect. 9 December, 2010
- There will be two separate finals for the two sections of 18.100B/C so make sure you do the right one! I will post some practice questions later today. 7 December, 2010.
- Solutions to HW10 posted, 6 December, 2010.
- In response to demand I added a histogram of the grades for the Second Test to the solution file, 1 December, 2010.
- Yet another revised version of HW10 -- with a comment added to Q2 to the effect that completeness of Y is not needed here (it was needed in Q1).
- My solutions (not proofread, I hope they are not too bad) up. 1 December, 2010.
- Another small revision to HW10 -- wording in Question 3. 30 November, 2010.
- Online Subject Evaluations opened today, Tuesday, November 30. You have until Monday, December 13 at 9am to comment on the course. I am quite interested to hear what you have to say!
- Minor correction to Q1 on HW10. Also, in the last question you may (and indeed I would suggest that you SHOULD) use the Stone-Weierstrass Theorem which I thought I would already have covered. 30 November, 2010.
- Corrected version of HW10 up, 27 November, 2010
- Lecture 25, Thursday 9 December. Fundamental Theorem of Algebra. Review.
- Lecture 24, Tuesday 7 December. Stone-Weierstrass completed then Power series, exponential, logarithm and trigonometric functions.
- Lecture 23, Thursday 2 December. Peano-Cauchy and start on Stone-Weierstrass Theorem.
- Student evaluation opens on 30 November. I am interested to hear what you have to say about the course, including how much you felt you learnt, how we could help a bit more in the process etc. Please try to take the time to fill in the on-line form.
- Lecture 22, Tuesday 30 November. Ascoli-Arzelà Theorem Notes. Peano's theorem on existence of solutions to ordinary differential equations
- 21 November, 2010. Solutions to HW8 and HW9 posted.
- I have always learnt something from teaching this course! This time I realized that the intersection of a decreasing sequence of compact and connected sets is non-empty, compact (I knew that) and connected. This latter fact -- maybe you can find a better proof than me -- turned out to be something I wanted for the analysis of `generalized blow-down maps'; 20 November, 2010.
- Lecture 21, Thursday 23 November. Uniform convergence and differentiation. A continuous function which is nowhwere differentiable. Uniformly convergent sequences on a compact metric space are equicontinuous.
- Test 2 -- take home -- put up 18 November, 2010.
- HW10 up -- for those who want to get an early start on the last homework.
- Wording of two questions in HW9 refined. 17 November, 2010.
- Lecture 20. Thursday 18 November. Completeness of the metric space of continuous bounded functions. Uniform convergence and integrability.
- HW9 finally up, 13 November, 2010
- Lecture 19. Tuesday 16 November. Uniform convergence and continuity.
- Solutions to HW7 posted, 6 November, 2010.
- Lecture 18, Tuesday 9 November. Properties of the integral. Fundamental Theorem of calculus (two versions).
- HW8 posted, 6 November, 2010.
- Correction of deadline for HW7 -- it is as usual at 11AM (paper)/5PM (email). Thanks Anne.
- Lecture 17, 4 November, 2010. Monotonic functions. Three theorems on integrability of functions (Rudin 6.9, 6.10 and 6.11).
- Lecture 16, 2 November, 2010. Partitions, upper and lower sums. Riemman-Stieltjes integrable functions. Refinement. Maybe integrability of a continuous function. Given an example of a Riemann integrable function on an interval which has integral zero but is not identically zero.
- Slightly changed verion of HW7, 31 October, 2010.
- HW6 solutions posted, 29 October, 2010.
- HW7 posted, 26 October, 2010.
- Reworded question 4 in HW6, 26 October, 2010.
- Lecture 15, 28 October 2010. Mean value theorem. l'Hopital's rule (finite cases), higher derivivatives, Taylor's theorem. Something to think about: If a function f is defined on the whole real line, is differentiable at each point and has a bounded derivative, show that it is uniformly continuous on the real line.
- Solutions to HW5 posted, 22 October, 2010.
- I added 5 more questions to HW6 -- not to be handed in, but for your amusement. They are about connected sets. It also occurred to me that someone might want the TeX file of the questions to avoid having to retype them, so that is up too. 21 October, 2010.
- Lecture 14, 26 October 2010. Uniform continuity. Monotonic functions. Derivative. Note that the fact that the derivative is defined as the limit of the difference quotient is really why the notion of a limit of a function at a limit point of its domain was introduced earlier by Rudin.
- Lecture 13, 21 October 2010. Composition of continuous maps. Continuous image of a compact set is compact. [Uniform continuity.] Continuous image of a connected set is connected. Try the manta: The continuous image of a convergent sequence is convergent, the continuous image of a compact set is compact, the continuous image of a connected set is connected. Work out an example of a map which is not continuous but has the property that the image of any compact subset is compact (hint, in a finite set any subset is compact).
- HW6 posted, 20 October, 2010.

Question 4 reworded, 26 October, 2010. - Histogram added to Test 1 solutions.
- More comments added to Test 1 solutions, including description of grading policy. 19 October, 2010. The median on the test was 45.
- Lecture 12, 19 October, 2010. Limit of a function at a limit point, continuity of a function. Equivalence of continuity to the inverse image of any open set being open.
- My solutions to the Test posted, 18 October, 2010.
- Second revision of HW5 posted -- typos in each of the questions except No 4. 17 October, 2010
- Revised version of HW5 posted -- with confusing typos fixed. 17 October, 2010.
- HW5 posted, a bit late. Thanks for the reminder Michael. 15 October, 2010.
- Further correction to Test-pre. 12:10, 11 October, 2010.
- Lecture 11 (14 October) is the IN-CLASS TEST.
- Lecture 10 (12 October) Comparison, root and ratio tests for convergence of series. Power series. Absolute convergence. I don't plan to cover `Summation by Parts', the `Multiplication' (really convolution) part of the next section or `Rearrangements' in lectures. These are all relatively elementary and are important in various applications but are not strictly necessary for the rest of the course. Still, I suggest that you read through them when you get a chance (!)
- Okay, I seemed to have survived the horrible cold I caught last week, and have now corrected the errors and confusion in the pre-test which have been reported so far! 11 October, 2010.
- Detailed schedule modified to be a bit closer to reality, 8 October, 2010.
- The conclusion, concerning Test 2, of discussions in class was that it will be a take-home exam with only limited time -- meaning you will have to work on it only in a single uninterupted period as decided (probably 2 hours). It will be made available Thursday 18 November and will have to be returned by Tuesday 23 November.
- Solutions to HW4 posted, 8 October, 2010.
- Date of second test corrected, again. Thanks Michael. I will poll people in lecture tomorrow, 7 Oct, to see whether a take-home test is the best solution.
- Did anyone notice I failed to post the solutions to HW3? Well, I have finally done so, 6 October, 2010.
- Homework schedule made saner by dropping one assignment -- Thanks to Harrison for pointing out the problem. (Changed again) 6 October, 2010.
- Test 1, `practice questions' put up, see below.

Note that the problems for the test will come from these. - Test 1, on 14 October. Will be in 4-237 and will be CLOSED BOOK -- no books, notespapers, computers, etc permitted. It will cover the material up to and including the `Cauchy sequences' section of Rudin -- so all of Chapters 1 and 2 and the first part of Chapter 3.
- Lecture 9 (7 October). Diameter of a set. Compact and Euclidean spaces are complete. Monotone sequences, liminf and limsup.
- Lecture 8 (5 October). Convergence of sequences, subsequences, Cauchy sequences, completeness, [Euclidean spaces are complete]. Convince yourself that a sequence in a metric space converges to a point iff either that point is the unique limit point of the image of the sequence or all but a finite number of terms of the sequence are equal to this point. Construct an example of a divergent (i.e. non-convergent) sequence in which every subsequence has a convergent subsequence (got it?)
- HW4 posted, 28 September, 2010.
- Hint added to last question in HW3.
- Lecture 7 (30 September). Proof that the closed interval [0,1] is not countable. Brief review of compactness. Connectedness. A connected metric space is one which
*cannot*be divided into two disjoint pieces, neither of which is empty, such that neither piece contains a limit point of the other. Try to convince yourself that [0,1] is connected but that the union of (0,1) and (1,2) is not connected. - Most marks lost in HW2 were for
- Not completing questions -- read the question again after you have finished to make sure you did not miss something.
- Some people did the first question for the real numbers.
- A small number of people tried to prove a set was open by proving it was not closed.
- HW2 solutions posted, 23 September, 2010.
- Lecture 6 (Tuesday 28 September). Heine-Borel-Weierstrass. The basic result is the compactness of the closed interval [0,1] in the real numbers. The argument goes as follows -- suppose there is an open cover with no finite subcover. Cut the interval into two closed `halves' then the same open cover of one of these has no finite subcover -- continue halving and choosing a `bad' half. Then take the sup of the bottoms of these bad intervals. The fact that this point -- the existence of which is a basic property of the reals -- is in one of the open sets is a contradiction since eventually the bad intervals are contained in this open set. Try it!
- After Lecture 5 -- if you found this really hard going, try to go through the proof of one theorem about compact sets carefully until you really understand it -- even if this takes a while. I would suggest looking at the Corollary following Theorem 2.36 in Rudin. Assume you know Theorem 2.34 -- that compact sets are closed -- and then try to prove the Corollary directly without using the preceeding theorem. The proof is the same of course, but with a countable collection rather than an arbitrary one and the added fact that they are getting smaller.
- 22 September, 2010. HW3 posted.
- Lecture 5 (Thursday 23 September). Compact sets. Read the definition and make sure you understand the quantifiers: Every open cover has a finite subcover. Try the proof of Theorem 2.35 yourself.
- HW1 was pretty well done so I don't have much to add except: Keep it up!
- 18 September, 2010: Oops, forgot to put up the solutions to HW1, but done now.
- 16 September, 2010: HW2 corrected
- Lecture 4 (Tuesday 21 September). The complement of an open set is closed and conversely. Topologies. Closure of a set. Maybe start compactness.

Suggestions: Make sure you go through the examples in 2.21. Make*absolutely sure*that you know an example of a subset of a metric space (e.g. the reals) that is*neither open nor closed*(to think that every subset of a metric space is either open or closed is a standard and very serious error). Show that for the discrete metric on any set, every subset is both open and closed. - HW2 posted, 15 Sept.
- Lecture 3 (Thursday 16 September). Countability of the Cartesian product of two countable sets. Metric spaces, Euclidean spaces, open sets, limit points.

The immediate aim is to understand open sets, limit points and closed sets in a metric space and the relation between them -- especially Theorem 2.23. - Lecture 2 (Tuesday 14 September). Existence of nth roots. Sets,
maps, in-, sur- and bijections, sequences. Countable and uncountable
sets. Definition of metric spaces.

Suggestions: Re(re)ad Theorem 1.21. Read Chapter 2 to Sect 2.16. Either before or after lecture, think about why the set of sequences of rationals which terminate at some point (are 0 from some point onwards) is countable and contrast this with Theorem 2.14. - 8 September, 2010. Homework 1 is up -- see below.
- 8 Sept:- Lecture 1 (Thursday 9 September)
I will cover much of the material in Rudin Chapter 1, please read this before lecture.

Suggestions. Go through the proof that a square-root of 2 cannot be rational but do it for the square-root of 7. Where does the proof fail for the square root of 4? Check that the natural numbers do have the least upper bound property and that the least upper bound of any non-empty set of natural numbers which is bounded above is in the set. Show that the numbers of the form a a√2+b where a and b are rational, is a field.