MAT 125B: Real Analysis (Winter 2015)

MAT 125B: Real Analysis (Winter 2015)

Homework Assignments

Each assignment is due by the end of lecture on the given date. Late homework will not be accepted. Solve the assigned problems in order, staple the properly assembled pages, and print your name clearly on the front page and write the number of your section (either A1 or A2) in red on the top right corner of the front page. Do not fold. Your assignment will be graded and then then returned to you in the discussion session. Assignments will be graded not only on correctness but also on the quality of writing; in particular, the reader is instructed to stop reading if a solution is incoherent.

The problems listed below are, unless stated otherwise, from Wade's book. Click on the link with the section number if you do not have the book yet. Those are the problems you need to turn in (again, unless explicitly stated otherwise). Some solutions will be provided, but will not be carefully proofread, so check for mistakes!

Due Date Problems
HW1 Wed., Jan. 14 5.1: 0, 7, 9

Note. Skip Problem 5.1.9 this time; it will be assigned later when we have better methods to solve it.

Solutions.

HW2 Wed., Jan. 21 5.1: 9

5.2: 0(a,b,d), 8, 10.

Note. Those two functions in 5.2.10 are max(f,g) and min(f,g). Prove first that max(f,g)=(|f-g|+f+g)/2 and min(f,g)=f+g-max(f,g).

Solutions.

HW3 Wed., Jan. 28 5.3: 0(a,b,c), 2(b,c), 4(b), 6.

Note. In the book's definition, a function f is called increasing if it is nondecreasing (that is, if a<=b implies f(a)<=f(b)).

Solutions.

HW4 Wed., Feb. 4 5.4: 0(c,d), 1(c), 2(d), 4(e), 8

7.1: 2(a,c)

Solutions.

HW5 Wed., Feb. 11 9.3: 2(a,b), 3(a), 5

9.4: 6, 8

Note. In problem 9.3.2, determine whether the limit exists and, if it does, compute it. Ignore the iterated limits, which we will not discuss. In problem 9.4.8, prove first that a uniformly continuous function maps Cauchy sequences into Cauchy sequences, and that every Cauchy sequence in Rn is convergent. This gives you a way to define the function g on E.

Solutions.

HW6 Wed., Feb. 18 11.1: 2(a,b), 3

Solutions.

HW7 Wed., Feb. 25 11.2: 2, 3, 4, 5, 7

Solutions.

HW8 Wed., Mar. 4 11.4: 2(b), 4

11.5: 6

11.6: 1(c)

Note. In problem 11.6.1(c), f-1 is not uniquely defined, as there is more than one point (u,v) that maps into (a,b), and each choice gives a different inverse. To remove the ambiguity, restrict to choices of (u,v) for which 0<u<v.

Solutions.

No more homework! HW8 was the last.