MATH 4433 - Introduction to Analysis I, Section 001 - Spring 2007
MWF 9:30-10:20 a.m., 120 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.

Office Hours: Mon 2:30-3:30 p.m., Wed 10:30-11:30 a.m., Thu 2:30-3:30 p.m., or by appointment.

The final exam will be on Friday, May 11, 8:00-10:00 a.m.

Prerequisite: MATH 2513 (Discrete Mathematical Structures) or permission of the instructor.

Course catalog description: Review of real number system. Sequences of real numbers. Topology of the real line. Continuity and differentiation of functions of a single variable. (F, Sp, Su)

Text: Steven R. Lay. Analysis With an Introduction to Proof, 4th edition, Pearson/Prentice Hall, 2005, ISBN 0-13-148101-0. The course will cover major parts of chapters 1-6.

Homework (solutions are deposited after the due date in the Chemistry-Mathematics Library, 207 PHSC):

• Homework 1, due Fri, Jan 26.
• Homework 2, due Fri, Feb 2.
• Homework 3, due Fri, Feb 9.
• Homework 4, due Fri, Feb 16.
• Homework 5, due Fri, Mar 2.
• Homework 6, due Fri, Mar 9.
• Homework 7, due Fri, Mar 16.
• Homework 8, due Fri, Apr 6.
• Homework 9, due Fri, Apr 13.
• Homework 10, due Fri, Apr 20.
• Homework 11, due Wed, Apr 25. Note the unusual due date!

Course content:

• Logic and proof: logical connectives, quantifiers, techniques of proof.
• Sets and functions: set operations, relations, functions, cardinality.
• The real numbers: natural numbers and induction, ordering, the completeness axiom, topology of the reals, compact sets, metric spaces.
• Sequences: convergence, limit theorems, monotone sequences, Cauchy sequences, subsequences.
• Limits and continuity: limits of functions, continuous functions, uniform continuity.
• Differentiation: derivative, mean value theorem, L'Hospital rule, Taylor's theorem.

Content of the lectures:

• Lecture 1 (Wed, Jan 17): Logical connectives: statement, truth value, examples, connectives, negation ∼p, conjunction p∧q (Sec. 1).
• Lecture 2 (Fri, Jan 19): Logical connectives (cont.): disjunction p∨q, implication (conditional statement) p⇒q, different ways to say p⇒q, equivalence p⇔q, tautology, examples. (Sec. 1).
  Quantifiers: universal quantifier ∀, existential quantifier ∃, s.t. ("such that"), truth values of the statements [∀x, p(x)] and [∃x s.t. p(x)] for p(x):x²-x-2=0; examples of mathematical shorthand (Sec. 2).
• Lecture 3 (Mon, Jan 22): Quantifiers (cont.): negating quantifiers, examples (Sec. 2).
  Techniques of proof I: inductive reasoning, counterexamples, deductive reasoning; contrapositive (∼q)⇒(∼p), converse q⇒p, and inverse ((∼p)⇒(∼q)) of the implication p⇒q; equivalence of an implication and its contrapositive (Sec. 3).
• Lecture 4 (Wed, Jan 24): Techniques of proof II: strategies:
  • direct proof of p⇒q;
  • proof of the contrapositive statement, (∼q)⇒(∼p), of the statement p⇒q;
  • coming to a contradiction c:
    • using that (∼p)⇒c is equivalent to p,
    • using that p∧(∼q)⇒c is equivalent to p⇒q;
  • by cases, which uses that (p∨q)⇒r is equivalent to (p⇒r)∧(q⇒r);
  • by induction (to be covered later)
  (Sec. 4).
• Lecture 5 (Fri, Jan 26): Basic set operations: sets and elements, a∈A, a∉A; subsets B⊆A, proper subsets B⊂A, equal sets B=A; sets of natural numbers, integers, rational numbers, real numbers; interval sets [a,b], (a,b], [a,b), (a,b); empty set ∅; union B∪A, intersection B∩A, complement B\A of B in A, universal set of U, complement A^c (of A in U); Venn diagrams, elementary properties of set operations, indexed families of sets (Sec. 5).
• Lecture 6 (Mon, Jan 29): Relations: (skip the part "Ordered pairs") Cartesian product (cross product) A×B of sets A and B, a relation R⊆A×B between A and B, a relation R⊆A×A on A, equivalence relations, equivalence classes, proof that two equivalence classes are either the same or are disjoint, a partition of a set; please read page 55 and do all practice problems (Sec. 6).
• Lecture 7 (Wed, Jan 31): Functions: function, domain, range, codomain; surjective ("onto"), injective ("one-to-one"), and bijecive functions (pages 60-62 of Sec. 7).
• Lecture 8 (Fri, Feb 2): Functions (cont.): characteristic function χ_A of a subset A⊆S (other notations: 1_A, I_A); functions acting on sets; compositions of functions; inverse functions, identity function i_A on A, inverse of a composition of functions, examples (pages 63-71 of Sec. 7).
  Please, do all practice problems, think how to prove each theorem before you read its proof from the book, and find examples of functions f:A→B such that the inclusions C⊆f^-1(f(C)), f(f^-1(D))⊆D, f(C₁∩C₂)⊆f(C₁)∩f(C₂) from Theorem 7.15 are strict).
• Lecture 9 (Mon, Feb 5): Cardinality: equinumerous sets, finite and infinite sets, finite and transfinite cardinal numbers (pages 77-78 of Sec. 8).
• Lecture 10 (Wed, Feb 7): Cardinality (cont.): denumerable sets, countable (=finite or denumerable) sets, uncountable sets, infinite sets (=denumerable or uncountable) sets; theorems:
  "Theorem": N is countable.
  Theorem: Z is countable.
  Theorem: If T is countable and S⊆T, then S is countable (without proof!).
  Theorem: If A and B are countable, then A×B is countable.
  Theorem: Q is countable.
  Theorem: If A and B are countable, then A∩B is countable (do the proof yourself).
  Theorem: A countable union of countable sets is a countable set (try to do the proof yourself; also look at Example 8.11(d) on page 83).
  Theorem: R is uncountable (Theorem 8.12).
  (pages 79-84 of Sec. 8; do not read the proofs of Theorems 8.9 and 8.10 and Example 8.11(a,b,c)).
• Lecture 11 (Fri, Feb 9): Natuaral numbers and induction: well-ordering property of N, principle of mathematical induction, example (Sec. 10).
• Lecture 12 (Mon, Feb 12): Ordered fields: axioms of addition, multiplication, distributive law, and axioms of order, elementary consequences of the axioms (Theorem 11.1) (pages 108-111 of Sec. 11).
• Lecture 13 (Wed, Feb 14): Ordered fields (cont.): examples of ordered fields (R and Q); the irrational numbers do not form a field (why?), Theorem 11.7 (that x≤y+ε for every ε>0 implies x≤y) (pages 111-113 of Sec. 11).
• Lecture 14 (Fri, Feb 16): Ordered fields (cont.): absolute value, properties, triangle inequality (pages 113-114 of Sec. 11).
  The Completeness Axiom: "holes" in Q, irrationality of numbers of the form p^1/2 for prime p (do the proof yourself), bounded above sets, bounded below sets, bounded sets, upper bound and lower bound, maximum and minimum, supremum and infimum, the Completeness Axiom, Theorem 12.7 (pages 117-121 of Sec. 12).
• Lecture 15 (Mon, Feb 19): Hour exam 1.
• Lecture 16 (Wed, Feb 21): The Completeness Axiom (cont.): Theorem 12.7 (finish the proof) (pages 121-122 of Sec. 12).
• Lecture 17 (Fri, Feb 23): The Completeness Axiom (cont.): Theorem 12.8; The Archimedean Property of R, Theorem 12.10 (do the proof yourself!), Theorem 12.11 (proof optional); Theorem 12.12 (density of Q in R), Theorem 12.14 (do the proof yourself!) (pages 122-126 of Sec. 12).
• Lecture 18 (Mon, Feb 26): Topology of the reals: ε-neighborhood N(x,ε) of x, deleted ε-neighborhood N^*(x,ε) of x, interior and boundary points of S⊆R, intS and bdS; closed and open sets, (S open) iff (S=intS), (S closed) iff (S^c=R\S open) (pages 129-131 of Sec. 13).
• Lecture 19 (Wed, Feb 28): Topology of the reals (cont.): Theorem 13.10 (the union of any collection of open sets is open, the intersection of any finite collection of open sets is open); Theorem 13.11 (the intersection of any collection of open sets is open, the union of any finite collection of open sets is open); examples and counterexamples; accumulation points and isolated points of sets, S', closure clS of S, Theorem 13.17 (do the proof yourself) (pages 132-134 of Sec. 13).
• Lecture 20 (Fri, Mar 2): Compact sets: open cover, subcover, compact sets, examples (pages 138-139 of Sec. 14).
• Lecture 21 (Mon, Mar 5): Compact sets (cont.): Heine-Borel's Theorem (proof of the fact that S compact ⇒ S closed and bounded; the proof of the converse is optional), Bolzano-Weierstrass Theorem (pages 139-142 of Sec. 14).
• Lecture 22 (Wed, Mar 7): Compact sets (cont.): Theorem 14.7 is optional (it is very powerful if the family contains uncountably many compact sets), Nested Intervals Theorem (different proof from the one in the book), (pages 142-143 of Sec. 14).
  Convergence: sequences, examples, convergent sequences, limit, examples (read everything from pages 156-160 before Theorem 16.8).
• Lecture 23 (Fri, Mar 9): Convergence (cont.): Theorem 16.8, examples; boundedness of a sequence (s_n), Theorem 16.13 (boundedness of convergent sequences), Theorem 16.14 (uniqueness of limit of convergent sequences) (pages 160-163 of Sec. 16).
  Limit theorems: Theorem 17.1 - limits of (s_n+t_n), (ks_n), (k+s_n), (s_nt_n), (s_n/t_n) (pages 165-166 of Sec. 17).
• Lecture 24 (Mon, Mar 12): Limit theorems (cont.): Theorem 17.1(d) - limit of (s_n/t_n), examples; Theorem 17.4 (s_n≤t_n ∀n∈N implies s≤t), Theorem 17.7 ("ratio test"); infinite limits, Theorem 17.12, Theorem 17.13 (pages 167-171 of Sec. 17 - read everyting and do all the proofs).
• Lecture 25 (Wed, Mar 14): Monotone sequences and Cauchy sequences: increasing, decreasing, monotone sequences, Monotone Convergence Theorem, example - Newton method for finding x^1/2 (pages 174-176 of Section 18, including Examples 18.4-5 and Practice 18.6-7).
• Lecture 26 (Fri, Mar 16): Monotone sequences and Cauchy sequences (cont.): Theorem 18.8 (limits of unbounded increasing/decreasing sequences); Cauchy sequences, every convergenct sequence is Cauchy, Cauchy convergence criterion, examples (pages 176-179 of Section 18 - read all proofs and examples, do all practice problems and the problems with hints or solutions given in the back of the book).
• Lecture 27 (Mon, Mar 26): Subsequences: subsequences - definition and examples, Theorem 19.4 (all subsequences of a convergent sequence converge to the same limit), practical uses of this theorem (pages 181-182 of Section 19).
• Lecture 28 (Wed, Mar 28): Hour exam 2.
• Lecture 29 (Fri, Mar 30): Subsequences (cont.): Theorem 19.7 (every bounded sequence has a convergent subsequence), Theorem 19.8 (every unbounded sequence has a subsequence that tends to +∞ of -∞); definition of the number e as a limit of a sequence, proof of convergence of this limit (pages 183-184 of Section 19).
• Lecture 30 (Mon, Apr 2): Limits of functions: definition of a limit of a function, examples (pages 190-192 of Sec. 20).
• Lecture 31 (Wed, Apr 4): Limits of functions (cont.): sequential criterion for limits, example of application, limit of a multiple kf of a function f, limits of a sum f+g, product fg and quotient f/g of two functions f and g, applications, one-sided limits (pages 193-196 of Sec. 20).
• Lecture 32 (Fri, Apr 6): Continuous functions: definition of a continuous function, equivalent definitions of continuity, relation with limits of functions, examples (pages 199-202 of Sec. 21).
• Lecture 33 (Mon, Apr 9): Continuous functions (cont.): proving that a function is discontinuous at a point by using sequences (Theorem 21.6), examples (Dirichlet funcion and modified Dirichlet function), continuity of a sum, product and ratio of continuous functions, continuity of a composition of continuous functions, examples (pages 202-205 of Sec. 21).
• Lecture 34 (Wed, Apr 11): Continuous functions (cont.): Theorem 21.14 (criterion for continuity based on preimages of open sets), examples (pages 205-206 of Sec. 21).
  Properties of continuous functions: bounded functions, compactness of the image of a compact set under a continuous functions (Theorem 22.2), corollary (continuous functions on a compact set reach their minimum and maxinum values), examples and counterexamples (pages 209-210 of Sec. 22).
• Lecture 35 (Fri, Apr 13): Properties of continuous functions: positiveness of f(x) for a continuous function in a neighborhood of a point c if f(c)>0 (and negativeness of f(x) if f(c)< 0), existence of a root of f(x)=0 in (a,b) if f(a)<0<f(b), Intermediate Value Theorem, examples of aplications (pages 210-213 of Sec. 22).
• Lecture 36 (Mon, Apr 16): The derivative: definition of the derivative of a function, examples, differentiability and continuity, derivatives of kf, f+g, fg, f/g (pages 231-236 of Sec. 25).
• Lecture 37 (Wed, Apr 18): The derivative (cont.): derivatives of xⁿ for natural n (using the product rule) and for all n (using the quotient rule), Chain Rule (Theorem 25.10) (pages 236-238 of Sec. 25).
  The Mean Value Theorem: Theorem 26.1 (vanishing of f' at local extrema), Rolle's Theorem (pages 241-243 of Sec. 26).
• Lecture 38 (Fri, Apr 20): The Mean Value Theorem (cont.): Mean Value Theorem, example of application (Bernoulli's inequality), f'(x)=0 for all x∈[a,b] as a sufficient condition of constancy of all f on [a,b], strictly increasing and strictly decreasing functions, conditions for f to be strictly increasing or strictly decreasing in terms of f' (pages 243-245 of Sec. 26).
• Lecture 39 (Mon, Apr 23): The Mean Value Theorem (cont.): Intermediate Value Theorem for Derivatives (compare and contrast with the Intermediate Value Theorem for Functions - Theorem 22.6; look also at Problem 25.6 on page 240); Inverse Function Theorem (pages 245-247 of Sec. 26).
• Lecture 40 (Wed, Apr 25): L'Hospital's Rule: Cauchy Mean Value Theorem, L'Hospital's Rule - motivation and examples (pages 251-253 of Sec. 27).
• Lecture 41 (Fri, Apr 27): Hour Exam 3.
• Lecture 42 (Mon, Apr 30): L'Hospital's Rule: proof of the theorem (page 253 of Sec. 27).
• Lecture 43 (Wed, May 2): Taylor's Theorem: statement and proof of Taylor's theorem (pages 259-261 of Sec. 28).
• Lecture 44 (Fri, May 4): Taylor's Theorem: applications of the theorem, estimates of the error terms, basic Taylor polynomials, cautionary remarks about the convergence of the Taylor series to the function's value, analytic functions (pages 261-25 of Sec. 28).

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or phone call from a doctor or a parent). You should come to class on time; if you miss a quiz because you came late, you won't be able to make up for it.

Homework: It is absolutely essential to solve a large number of problems on a regular basis! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. All hand-in assignments must be submitted in class on the due date. No late homework will be accepted!

You are encouraged to discuss the homework problems with other students. However, you have to write your solutions clearly and in your own words - this is the only way to achieve real understanding! It is advisable that you first write a draft of the solutions and then copy them neatly. Please write the problems in the same order in which they are given in the assignment.

Shortly after a homework assignment's due date, solutions to the problems from that assignment will be placed on restricted reserve in the Chemistry-Mathematics Library in 207 PHSC.

Quizzes: Short pop-quizzes will be given in class at random times; your lowest quiz grade will be dropped. Often the quizzes will use material that has been covered very recently (even in the previous lecture), so you have to make every effort to keep up with the material and to study the corresponding sections from the book right after they have been covered in class.

Grading: Your grade will be determined by your performance on the following coursework:

• three in-class exams, 50 minutes each (each midterm contributes 15% to your overall grade);
• homework (15% of your overall grade; the lowest homework grade will be dropped);
• pop-quizzes (10% of your overall grade; the lowest quiz grade will be dropped);
• final exam (30% of your overall grade).

Academic calendar for Spring 2007.

Policy on W/I Grades : Through February 23, you can withdraw from the course with an automatic W. In addition, it is my policy to give any student a W grade, regardless of his/her performance in the course, through the extended drop period that ends on May 4. However, after April 2, you can only drop via petition to the Dean of your college. Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.

The grade of I (Incomplete) is not intended to serve as a benign substitute for the grade of F. I only give the I grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it! For more details on the University's policies concerning academic misconduct see http://www.ou.edu/provost/integrity/.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD (only) 405-325-4173.

Use of calculators and technology: Because of the nature of the course, calculators are not needed (and will not be allowed during the exams).

Good to know:

• The Greek_alphabet.