## Math 5201 Real Variables

### Written Assignments and Tests

Text: Real Mathematical Analysis, by Charles C. Pugh, Springer, 2002. (or paperback version 2010, or second edition 2015)

• Set 1 Due Friday Sept. 2:
• Review your (or any) 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 x2-3 = 1. Use an epsilon-delta argument.
• C. Prove: The function f defined by f(x)=x2-3 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^2-3) = 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 12 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. 16:
• 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 (prove one direction only; state clearly which direction you are proving!), , 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).
• MOVED TO HW SET 3: Related to 39: Prove that sin(1/x) is not uniformly continuous on (0,1).
• Extra Credit: 35a.

• Quiz 2: Date Mon. Sept 26. 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 Fri. Sept 30:
• Ch 1, Related to 39: Prove that sin(1/x) is not uniformly continuous on (0,1).
• Read Sections 2.1 - 2.2.
• Ch 2: 2,3, 6a,7,12 (w/o "more generally"), 25a, 26

• HW 1 corrections due Wednesday Sept. 28.
• Midterm 1: Wednesday, Oct. 5, 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 .

HW Set 2 corrections Due Friday Oct. 7 (due date moved to Monday Oct. 10)
• Quiz 3: Wednesday Oct. 12 - Chart to fill in like in Test 1. Decide whether a subset A of a metric space is open, closed, compact. Determine Cl(A), int(A), Lim(A), A' (cluster points of A).
• Set 4 Due Mon. Oct. 17:
• Read Sections 2.1 - 2.2.
• Ch 2: 8,22,23,39,89a.
• Extra Credit: 46. Due with HW 5, but announced now.

• Quiz 4: Monday Oct. 24. 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 Wed. Nov. 2:
• 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.

• Fri. Nov. 4 NO CLASS
• Quiz 5: Wed. Nov. 9. Tentative list of 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).)
Work on HW corrections and prepare for Test 2.
• Midterm 2: Wed. Nov. 16. 3-4:30. Mostly on Chapter 2 in Pugh. Test 2 topic list.
• Fri. Nov 18 NO Class.
• Thanksgiving Break Nov. 24-25
• Set 6 Due Monday Nov. 28:
• 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.

• Quiz 5: Friday Dec. 2. 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. 7:
• 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.

• Last day of class: Friday Dec. 9. HW corrections due unless later date is arranged.

• Final Exam. Wed. Dec. 14, 10-11:55PM. Same format as midterms 1 and 2, but cumulative. Ch's 1-5. See midterm 1 and 2 topics lists above. Add to them the following:
1. Know the 4 definitions for a subset A of M to be dense and proof that any one implies any other. ("Closure =", "closure contains", neighborhood, sequence)
2. Know the definition of cuts and the reason for introducing them.
3. Know the definition for a Lipschitz function and a contraction mapping.
4. Prove that if a sequence of continuous functions converges uniformly, then the limit function is also continuous.
5. Prove that sequences generated by a contraction mapping are Cauchy.
6. Compute one or more Picard iterates for approximate solutions to an ODE.
7. Determine the C0 (as a subspace of Cb) and/or L^1 distance between given functions.
8. Determine whether a sequence of functions converges pointwise, in C^0, or in L^1.