# Math 131BH Homework Assignments

General remarks:
• You are permitted to use all resources available to you (textbook, notes, other students, TA, office hours, virtual office hours, etc.) to do these homework assignments.  However, you will get the most benefit from the homework if you first attempt to do all the questions yourself, without outside assistance.  Only after you have tried a question can one truly appreciate the solution.  If you find yourself relying more and more on external help in order to complete each week's homework, then this is a danger sign and may indicate that you will have significant trouble with the midterms and final.  Copying your homework from someone else without first trying it yourself may help you score better on the 15% of the grade that comes from the homework, but may at the same time cheat you out of the 85% of the grade that comes from the exams.  (Trying to make up for this later by reading other people's solutions to the homework may not necessarily be effective.  By far the best way to learn is to do the homework yourself, without assistance).  The homework assignments are definitely long, tough, and time-consuming - this is an Honors course, after all - but they will help you understand the course material far more thoroughly, and the exams will be much easier to handle if you took the time to do the homework by yourself.  (You will also get a lot more out of the discussion section if you have already tried all the homework problems beforehand).
• In your proofs, you may use all knowledge available to you, whether from previous classes (in particular, math 131AH), or from other mathematics books, etc.  Since the material covered here is somewhat more advanced than in Math 131AH, I will not expect you to give completely rigorous proofs that give detail all the way down to the basic axioms; however I do expect your proofs to be clear, correct, and relevant to the question.  (For instance: if asked to prove that X implies Y, you should not devote your entire answer to proving that Y implies X; if asked to prove that for every epsilon there exists a delta which obeys property P, you should not devote your answer to showing that for every delta and every epsilon the property P holds, etc.)

Assignment 1 (Due Apr 11)

Assignment 2 (Due Apr 18)

Assignment 3 (Due Apr 25)
• Available in PDF format here.
• Errata: in Q4 (i.e. in the proof of Proposition 4), it should be stated as a hypothesis that Y is complete. (Otherwise the proposition is false). In Q6(b), "Proposition 5" should instead read "Proposition 6", and the first part of the hint should be ignored.

Assignment 4 (Due May 2)
• Available in PDF format here.
• Errata: In Q1 of Assignment 4, in the sketch of proof given in page 2 of Week 4/5 notes, the integrals of f_n on [x,x_0] and [x_0,x] should instead read f'_n. In Q3, all occurrences of "8" in the problem should be replaced by "32".

Assignment 5 (Due May 9)
• Available in PDF format here.
• Errata: in Q9, Theorem 18 should refer instead to Theorem 19. In Q7, part (b) is extremely difficult and should be dropped. (Part (c) should then also be dropped, since it depends on (b)).

Assignment 6 (Due May 16)
• Available in PDF format here.
• Errata: In Q8, all occurrences of nx should instead read 2 pi nx. In Q4, C(R;Z) should read C(R/Z;C). In Q7, e^{pi i N x} should instead read e^{pi i (N-1) x}.

Assignment 7 (Due May 23)

Assignment 8 (Due Jun 6)
• Available in PDF format here.
• Errata: in Q1(b), one needs the additional assumption that m(A_1) is finite. Also in Q1, the limits should be indexed by j, not n. In the definitions of boxes in Q2, x_j should range between 0 and 1/q, not 0 and 1/m. In Q5-8, all occurrences of "mu" should read "m" instead. In Q7(b), A should be an open box in R^n, not in R. In Q9(b), m(A/B) = m(A) - m(B) should instead read m(B/A) = m(B) - m(A).

Assignment 9 (Due Jun 6)
• Available in PDF format here.
• Errata: In Q8, Corollary 4 should instead refer to Corollary 5. In Q10(b), in part (b) of Proposition 7, the right-hand side should read "integral_Omega f + integral_Omega g" and not just "integral_Omega g".