Math 5201 Real Variables
Fall 2013
Prof. Peckham
Homeworks and Tests
Text: Real Mathematical Analysis,
by Charles C. Pugh, Springer, 2002.
- Set 1 Due Wednesday Sept. 11:
- Review your calculus text for definitions of the limit of function, continuity, derivative, and
the limit of a sequence.
- A. Prove: lim as x->2 of x2 = 4.
- B. Prove: The function F defined by f(x)=x2 is continuous at x=2.
- C. Prove: The derivative of f(x)=x2 at x=2 is 4.
- D. Prove: lim as n->infinity 1/(n^2+1) = 0.
- Read: Sections 1.1-1.2. In 1.2, skim pp 13 from the Pf of Theorem 2 through p. 16. Skim the rest of Chapter 1.
- Written HW: Ch 1: 2a-c,3a,5a,7d. Extra credit: 8.
- Quiz 1: Wednesday Sept. 18. (Move to Fri Sept 20.) at end of class.
Prove 1. The monotone covergence theorem for nondecreasing sequences bounded above
2. One of the following series theorems: Basic comparison test, limit comparison test, ratio test.
Replace: LCT with integral test and add the proof of the convergence of geometric series.
- Set 2 Due Wednesday Sept. 25 (extended to Fri. Sept. 27):
- Read Section 1.4. Look through 1.5 - 1.6.
- Show the sequence defined by an = log(n) is not Cauchy, but does satisfy
|an - an+1|->0.
Use espsilon-N definition directly. Do not directly use the fact that the sequence is known to be unbounded or that the sequence is known to diverge.
- Show that if f:[a,b]->R, and f is nondecreasing, the one-sided limits exist for all x in (a,b).
- Written HW: Ch 1: 11, 17, 35b ONLY for empty or finite sets, 36a, 39, 42
Change 42d to the following easier questions:
- Find sequences such that limsup (a_n+b_n) < limsup (a_n) + limsup (b_n)
- Find a sequence and a constant c such that limsup(c a_n) is NOT equal to c limsup(a_n).
Extra Credit: 35a.
- Quiz 2: Friday Oct. 4.
Possible questions: Prove the rationals are countable. Prove the reals are not countable. Show the epsilon-delta version of continuity is equivalent to the sequence version of continuity. Compute limsup, liminf of specific sequences. Give examples of series where various tests (eg ratio) fail. More may be added.
- Set 3 Due Monday Oct. 7:
- Read Sections 2.1 - 2.2.
- Ch 2: 2,3, 6b,7,12 (w/o "more generally"), 25a, 26
- Extra credit: 30b
- HW 1 corrections due Wednesday Oct. 9.
- Midterm 1: Friday, Oct. 11, 3-4:30.
Sequences and series, topics covered in Chapter 1 with definition of cuts, Chapter 2: 2.1.
See Topic list below (TBA). "Basic theorems" and proofs.
Definitions and short problems. Examples.
Test 1 topic list .
Sample test will be handed out in class.
- HW Set 3 corrections, Test 1 corrections and Correction for "LUB prop of R" implies "GLB prop of R" from HW 2.
Due Monday Oct. 21.
- Set 4 Due Monday Oct. 28:
- Read Sections 2.1 - 2.2.
- Ch 2: 8,22,23,39,89a.
- Extra Credit: 46. Due at a time to be determined later, but announced now.
Dates below here to be confirmed.
- Quiz 3: Friday Nov. 1.
For this quiz, compact means sequentially compact.
Tentative list of possible questions:
(1) Prove A compact implies A closed. (2) Prove A compact implies A bounded. (3) Prove A compact and f continuous implies f(A) is compact. (4) Prove that a closed subset of a compact set is compact. (5) Prove that a sequence that converges in a product space using any of {Euclidean, Sum, max} metrics also converges in either of the other two metrics. (8) Examples: f continuous, A closed, f(A) not closed; f continuous, A bounded, f(A) not bounded; f bijection, continuous, f inverse not continuous.
- Set 5 Due Wed. Nov. 6:
- Read Sections 2.3 - 2.4.
- Ch 2: 27a, 47, 48, 49, 51, 54, 55, 91cdf, 92e AND give an example with strict inclusion.
- Extra Credit: 46.
- Quiz 4: Wednesday Nov. 13.
Possible questions: (1) Prove A covering compact implies A closed. (2) Prove A covering compact implies A bounded and/or totally bounded. (3) Prove that A totally bounded implies A is bounded. (4) Prove A covering compact and f continuous implies f(A) is covering compact. (5) Prove that a closed subset of a covering compact set is covering compact. (6) Are pairs of sets homeomorphic? Why or why not? (7) Examples: A connected, int(A) not connected; A disconnected, int(A) connected; A disconnected, cl(A) connected. 8) Prove that if (a_n) is a sequence in M and no subsequence of (a_n) converges to the point m in M, then there exists an r>0 for which a_n is in M_r m (the ball of radius r centered at m) for only finitely many n. (Hint: Prove the contrapositive: if no such r exists, then there will be at least one subsequence that converges to m).)
- Set 6 Due Monday Nov. 18:
- Read Ch 3: pp 139-142 (derivative), 146-149 (pathological functions)
- Ch 3: 14ab, 64a
For 14a, show only the first derivative at zero is zero and the first derivative
is continuous on the reals.
- HW Sets 4 and 5 corrections due Wednesday Nov. 20
- Midterm 2: Friday, Nov. 22. 3-4:30. Mostly on Chapter 2 in Pugh.
Test 2 topic list.
- Wed. Nov. 27 NO CLASS - replaced by extra time in earlier classes
- Fri. Nov. 29 NO CLASS - Thanksgiving Break
- Quiz 5: Friday Dec. 6.
Possible questions: 1. Proof of the uniform convergence theorem (Thm 1, Chapter 4). 2. Convergence of functions: pointwise, uniform, L1; examples which coverge pointwise, but not uniformly, or not in L1. 3. Examples of functions that are Ck but not Ck+1; differentiable, but not C1. More TBA???
- Set 7 Due Monday Dec. 9:
- Read Chapter 4, Section 1. Skim section 5.
- Ch 4: 5a-d
- Denote successive approximations via the Picard iteration method by yn(t). For the
initial value problem y'=y2, y(0) = 1, let y1(t)=1.
Compute y2 and y3 by hand, and y4 and y5
using Mathematica (or any other computational aid). Also compute the exact solution
using separation of variables. Rewrite the exact solution as a power series. Compare
your approximations y2, ..., y5 with the series solution.
- HW sets 2 and 6 corrections due Wed. Dec. 12
- HW set 7 corrections due Wed. Dec. 18 at the final exam.
- Final Exam Wed. Dec. 18, 10-11:55AM. Cumulative. Ch's 1-5.
See midterm 1 and 2 topics lists above. Add to them:
- Prove that if a sequence of continuous functions converges uniformly, then the limit
function is also continuous.
- Prove that a contraction mapping on a complete metric space has a unique fixed point.
- Compute one or more Picard iterates for approximate solutions to an ODE.
- Determine the C0 (as a subspace of Cb) and/or L^1 distance between given functions.
- Determine whether a sequence of functions converges pointwise, in C^0, or in L^1.
This page is maintained by
Bruce Peckham (bpeckham@d.umn.edu)
and was last modified on
Tuesday, 03-Dec-2013 16:32:28 CST.