Math 5201 Real Variables
Fall 2015
Prof. Peckham
Written Assignments and Tests
Text: Real Mathematical Analysis,
by Charles C. Pugh, Springer, 2002.
- Set 1 Due Friday Sept. 4:
- 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 3x+1 = 7. Use an epsilon-delta argument.
- B. Prove: lim as x->2 of x2-1 = 3. Use an epsilon-delta argument.
- C. Prove: The function f defined by f(x)=x2-1 is continuous at x=2. You may use your answer to part B.
- D. Prove: The derivative of f(x)=x2+1 at x=1 is 2. Compute directly from the definition of derivative, but you need not use epsilon-delta arguments.
- E. Prove: lim as n->infinity 1/(n^2+1) = 0. Use epsilon-N argument.
- Read: Chapter 1, sections 1-2. In sec 2, skim pp 13 from the Pf of Theorem 2 through p. 16. Skim the rest of Chapter 1. We will cover some of this material in more detail later.
- Textbook Assignment: Ch 1: 2a-c,3a,5a,7d. Extra credit: 8. Hints: For 2b, prove the first DeMorgan's law directly; prove the second law using the first law and complements. For 5a it would help to add a qualifying word such as "some" or "all".
- Quiz 1: Mon Sept 14 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, ratio test,
integral test, geometric series (for r<1).
- Set 2 Due Friday Sept. 18:
- Skim Section 1.3. Read Section 1.4. Look through Sections 1.5 - 1.6.
- Read Section 3.3 on Series, especially the comparison test, ratio test, and integral test.
- 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 limit of f(x) as x approaches b from the left exists.
Hint: mimic the ideas in the proof of MCT.
- Textbook assignment: Ch 1: 9a, 11 (hint: use contradiction), 17, , 42abce
- Related to 42d, but slightly easier:
- 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).
- Related to 39: Prove that sin(1/x) is not uniformly continuous on (0,1).
- Extra Credit: 35a.
- Quiz 2: Date Mon. Sept 28.
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.
- Set 3 Due Wed. Sept 30 Moved to Fri Oct 2:
- Read Sections 2.1 - 2.2.
- Ch 2: 2,3, 6a,7,12 (w/o "more generally"), 25a, 26
- Midterm 1: Monday, Oct. 5 Moved to Wed Oct 7, 3-4:30.
Sequences and series, topics covered in Chapter 1 with definition of cuts, Chapter 2: 2.1.
Basic theorems and proofs.
Definitions and short problems. Examples. See
Test 1 topic list .
See also Practice Test 1.
HW Set 1 corrections and HW Set 2 corrections Due Monday Oct. 12
- Set 4 Due Mon. Oct. 19:
- 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.
- Quiz 3: Monday Oct. 26.
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. (6) Examples: f continuous, A closed, f(A) not closed; f continuous, A bounded, f(A) not bounded; f bijection, continuous, f inverse not continuous; f discontinuous, sequence converging to a in domain, corresponding seq not converging to f(a) in range, open set with inverse image not open, closed set with inverse image not closed, epsilon for which there exists no delta that "works".
- Set 5 Due Mon. Nov. 2 (moved to Wed. Nov. 4):
- Read Sections 2.3 - 2.4.
- Ch 2: 47, 48, 49, 51, 54, 55, 91f, 92e AND give an example with strict inclusion.
- Extra Credit: 46.
- Quiz 4: Wed. Nov. 11.
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).)
- Midterm 2: Wed. Nov. 18. 3-4:30. Mostly on Chapter 2 in Pugh.
Test 2 topic list.
Test 2 from Fall 2013. .
- Set 6 Due Monday Nov. 23:
- Read Ch 3: pp 139-142 (derivative), 146-149 (pathological examples)
- Ch 3: 14ab, 64a
For 14a, show only the first derivative at zero is zero and the first derivative
is continuous on the reals.
Extra Credit: 14a. Show all derivatives at zero are zero.
- HW corrections through Set 5 due Wednesday Nov. 25
- Thanksgiving Break Nov. 26-27
- Quiz 5: Friday Dec. 4.
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.
- Set 7 Due Wednesday 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.
- Final Exam Mon. Dec. 14, 8-9:55PM. Same format as midterms 1 and 2, but cumulative. Ch's 1-5.
See midterm 1 and 2 topics lists above. Add to them:
- Know the 4 definitions for a subset A of M to be dense and proof that any one implies any other. ("Closure =" or "closure contains", neighborhood, sequence)
- Prove that if a sequence of continuous functions converges uniformly, then the limit
function is also continuous.
- Prove that sequences generated by a contraction mapping are Cauchy.
- 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
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and was last modified on
Sunday, 28-Aug-2016 16:46:22 CDT.