MATH 4433 - Introduction to Analysis I, Section 001 - Fall 2014
TR 10:30-11:45 a.m., 222 PHSC

Office Hours: M 2:30-3:30 p.m., T 1:30-2:30 p.m., or by appointment, in 802 PHSC.

Course catalog description: Prerequisite: 2433 and 2513 or permission of instructor. 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, Pearson, 5th ed., 2012, ISBN: 03217474X. The course will cover major parts of Ch. 1-6 (and, if time permits, parts of Ch. 7 and/or 8).

Homework [read the following useful advice on writing proofs and advice from students]

• Homework 1, due August 28 (Thursday).
• Homework 2, due September 4 (Thursday).
• Homework 3, due September 11 (Thursday).
• Homework 4, due September 23 (Tuesday) [final version (updated on September 16)]
• Homework 5, due October 2 (Thursday).
• Homework 6, due October 9 (Thursday).
• Homework 7, due October 16 (Thursday).
• Homework 8, due October 30 (Thursday).
• Homework 9, due November 6 (Thursday).
• Homework 10, due November 13 (Thursday).
• Homework 11, due November 20 (Thursday).
• Homework 12, due December 4 (Thursday).

Course content:

• Logic and proof: logical connectives, quantifiers, techniques of proof.
• Sets and functions: set operations, relations, functions, cardinality.
• The real numbers: natural numbers, induction, ordering, completeness, topology of R, compact sets.
• 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.
• (Tentative) Integration: Riemann integral, Fundamental Theorem of Calculus.

Content of the lectures:

• Lecture 1 (Tue, Aug 19): Logical connectives: statement, truth value, examples; connectives; negation ∼p; conjunction p∧q; disjunction p∨q; implication (conditional statement) p⇒q; different ways to say p⇒q; equivalence p⇔q; tautology, examples [Sec. 1.1]
• Lecture 2 (Thu, Aug 21): Logical connectives (cont.): [∼(p⇒q)]⇔[p∧(∼q)], more examples.
  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 negating quantifiers, examples [Sec. 2]
• Lecture 3 (Tue, Aug 26): 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. 1.3]
  Techniques of proof II: strategies:
  • direct proof of p⇒q (Example 1.4.1);
  • proof of the contrapositive statement, (∼q)⇒(∼p), of the statement p⇒q (example similar to Exercise 1.4.22: if x is irrational, then x^1/3 is irrational);
  • coming to a contradiction c:
    • using that (∼p)⇒c is equivalent to p (Exercise 1.4.15: log₂7 is irrational);
    • using that p∧(∼q)⇒c is equivalent to p⇒q (example will be given in Lecture 4).
  [pages 28-31 of Sec. 1.4]
• Lecture 4 (Thu, Aug 28): Techniques of proof II (cont.): strategies (cont.):
  • coming to a contradiction c:
    • using that (∼p)⇒c is equivalent to p (considered in Lecture 3);
    • using that p∧(∼q)⇒c is equivalent to p⇒q (proof that x>3 implies that x²−x−6>0);
  • by cases, which uses that (p∨q)⇒r is equivalent to (p⇒r)∧(q⇒r) (Example 1.4.5);
  • by induction (to be covered later).
  [pages 31-34 of Sec. 1.4]
  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 N, integers Z, rational numbers Q, real numbers R; interval sets [a,b], (a,b], [a,b), (a,b); empty set ∅; proof that ∅;⊆A for each set A [pages 38-42 of Sec. 2.1]
• Lecture 5 (Tue, Sep 2): Basic set operations (cont.): union B∪A, intersection B∩A, complement B\A of B in A, universal set U, complement A^c of A (in U); Venn diagrams; elementary properties of set operations; indexed families of sets [pages 42-47 of Sec. 2.1]
  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 [pages 52-56 of Sec. 2.2]
• Lecture 6 (Thu, Sep 4): Relations (cont.): equivalence classes, proof that two equivalence classes are either the same or are disjoint [page 56 of Sec. 2.2]
  Functions: a function from A to B as a relation ƒ⊆A×B satisfying certain properties; domain, codomain, range of a function; surjective ("onto"), injective ("one-to-one"), and bijecive functions; indicator (characteristic) function χ_S of a subset S⊆A (other notations: 1_S, I_S); visualizing functions (the graph of a function ƒ:A→B is the equivalence relation ƒ⊆A×B); the vertical line test in the light of the definition of ƒ as a relation; functions acting on sets: definitions of the image ƒ(C)⊆B of C⊆A and the pre-image ƒ⁻¹(D)⊆A of D⊆B; examples [pages 65-70 of Sec. 2.3]
  Reading assignment (mandatory): Theorem 2.3.16 and part of its proof (Practice 2.3.17); think of examples of functions ƒ:A→B such that the inclusions C⊆ƒ⁻¹(ƒ(C)), ƒ(ƒ⁻¹(D))⊆D, ƒ(C₁∩C₂)⊆ƒ(C₁)∩ƒ(C₂) from Theorem 2.3.16 are strict (i.e., the sets on the left are proper subsets of the sets on the right) [page 70 of Sec. 2.3]
  Reading assignment (optional): A partition of a set, properties of partitions [pages 57-58 of Sec. 2.2]
• Lecture 7 (Tue, Sep 9): Functions: conditions for the equalities C=ƒ⁻¹(ƒ(C)), ƒ(ƒ⁻¹(D))=D, ƒ(C₁∩C₂)=ƒ(C₁)∩ƒ(C₂) to hold (Theorem 2.3.18); compositions of functions; elementary properteis of composition (Theorem 2.3.20); inverse functions; identity function i_A on A; inverse of a composition of functions (Theorem 2.3.28); examples [pages 71-76 of Sec. 2.3]
  Cardinality: equinumerous sets A∼B; ∼ is an equivalence relation; finite and infinite sets; finite and transfinite cardinal numbers; denumerable sets (equinumerous with N); countable (=finite or denumerable) sets; uncountable sets [pages 82-84 of Sec. 2.4]
• Lecture 8 (Thu, Sep 11): Cardinality (cont.): a paradox about infinite sets: an infinite set can be equinumerous with its proper subset (example: the natural numbers N is equinumerous with the set A={1,4,9,16,25,...}⊆N of squares of natural numbers: the function ƒ:N→A defined by ƒ(n)=n² is a bijection); Principle of Well Ordering of N;
  Theorem 2.4.9 (with proof): If T is countable and S⊆T, then S is countable;
  Theorem 2.4.10 (without proof): Let S be nonempty. The following conditions are equivalent: (a) S is countable; (b) ∃ an injection ƒ:S→N ["seating every member s∈S in N seats if each seat can take only one s and some seats may be left empty"]; (c) ∃ a surjection g:N→S ["giving every member s∈S chocolote bars from a set of N chocolate bars, so that every s receives at least one bar"];
  more examples: N is countable, Z is countable, a countable union of countable sets is a countable set, Q is countable, the Cartesian product A×B of two countable sets A and B is countable (exercise!);
  Theorem 2.4.12: R is uncountable, with proof based on Georg Cantor's famous diagonal argument.
  [pages 85-89 of Sec. 2.4, page 105 of Sec. 3.1]
• Lecture 9 (Tue, Sep 16) Natural numbers and induction: Principle of Well Ordering of N; Principle of Mathematical Induction (with proof based on the Principle of Well Ordering of N), examples [pages 104-107 of Sec. 3.1]
  Ordered fields: axioms of ordered fields: axioms A1-A5 of addition, M1-M5 of multiplication, distributive law (LD), axioms O1-O4 of ordering; elementary consequences of the axioms of ordered fields (Theorem 3.2.2) [pages 113-116 of Sec. 3.2; read the proofs of all parts of Theorem 3.2.2, including Practice 3.2.3 and 3.2.4]
  Reading assignment: Statement of Theorem 3.1.6 (mandatory), proof of the theorem (optional) [pages 107-108 of Sec. 3.1]
  
• Lecture 10 (Thu, Sep 18): Exam 1 [on the material from Sections 1.1-1.4, 2.1-2.4 covered in Lectures 1-8]
• Lecture 11 (Tue, Sep 23): Ordered fields (cont.): Theorem 3.2.8 that (x≤y+ε ∀ε>0) ⇒ x≤y; absolute value; properties of absolute value (Theorem 3.2.10); a useful consequence of the triangle inequality: ||x|−|y||≤|x−y| [pages 118-110 of Sec. 3.2]
  The Completeness Axiom: "holes" in Q; irrationality of numbers of the form p^1/2 for prime p (do the proof yourself); sets bounded above, sets bounded below, bounded sets; upper bounds and lower bounds; maximum and minimum; supremum and infimum; the Completeness Axiom [pages 123-126 of Sec. 3.3]
  Reading assignment (mandatory): Proof of Theorem 3.2.10 [page 119, Sec. 3.2]
  Reading assignment (mandatory): Proof of Theorem 3.3.1 [pages 123-124 of Sec. 3.3]
  Reading assignment (mandatory): Statement and proof of Theorem 3.3.8 [page 127 of Sec. 3.3]
• Lecture 12 (Thu, Sep 25): The Completeness Axiom (cont.): Theorem 3.3.7 (supremum of the set C={x+y:x∈A,y∈B}); the Archimedean Property - N is unbounded above in R (Theorem 3.3.9); existence of a (positive) square root of any prime number (Theorem 3.3.12, only the statement!); density of Q in R: Lemma (Theorem 3.3.10(a)), Theorem 3.3.13 [pages 126-128, 130 of Sec. 3.3]
  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; examples; definition of closed and open sets; (S is open) iff (S=intS), (S closed) iff (S^c=R\S open) (Theorem 3.4.7); examples (Example 3.4.8, Practice 3.4.9) [pages 134-137 of Sec. 13]
  Reading assignment (mandatory): Theorem 3.3.15 [pages 130, 131 of Sec. 3.3]
• Lecture 13 (Tue, Sep 30): Topology of the reals (cont.): Theorem 3.4.10 (the union of any collection of open sets is open, the intersection of any finite collection of open sets is open); a "counterexample" - the intersection of collection {(-1/n,1/n):n∈N} of open sets is the closed set {0}; Theorem 3.4.11 (the intersection of any collection of closed sets is closed, the union of any finite collection of closed sets is closed) - think about the proof of this theorem and about a "counterexample", i.e., an example of a collection of closed sets whose union is not closed; accumulation points and isolated points of sets; set S' of accumulation points of S; closure clS of S; Theorem 3.4.17 (do the proof yourself) [pages 137-140 of Sec. 3.4]
  Compact sets: open cover, subcover, compact sets; examples - the open interval (0,2) is not compact, the set N is not compact, any finite subset of R is compact [pages 143-144 of Sec. 3.5]
• Lecture 14 (Thu, Oct 2): Compact sets (cont.): Lemma 3.5.4 that a nonempty closed bounded subset of R has a minimum and a maximum; Heine-Borel's Theorem that a set is compact iff if it closed and bounded (skip the proof); Bolzano-Weierstrass Theorem that a bounded subset of R that contains infinitely many points has an accumulation point (with proof); skip Theorem 3.5.7; Nested Intervals Theorem (with a proof different from the one in the book) [pages 145-148 of Sec. 3.5]
  Convergence: sequences, notations, examples; convergent sequences, divergent sequences, limit; examples, floor and ceiling functions [pages 161-165 of Sec. 4.1]
  Reading assignment (mandatory): Try to do the proof of Lemma 3.5.4 without looking in the book: write down the definitions of closed set, bounded set, minimum and maximum, then think how to connect them - i.e., a maximum of a set is a supremum of a set under certain condition, a bounded above set has a suprimum (which axiom was that?), a set is closed if it contains its boundary or, eqiuvalently, if it contains all its accumulation points, ...
  Reading assignment (mandatory): Read Example 4.1.6 carefully (reproducing all details in the calculations!), and do Practice 4.1.7 and Practice 4.1.10 [pages 165, 166 of Sec. 4.1]
• Lecture 15 (Tue, Oct 7): Convergence (cont.): more examples on proving convergence of sequences directly from the definition; if the "distances" |s_n−s| from s_n to s are bounded above by a convergent sequence, then lim(s_n)=s (Theorem 4.1.8); example of application (Example 4.1.9); proving that a sequence is divergent (Example 4.1.12); every convergent sequence is bounded (Theorem 4.1.13); if a sequence converges, its limit is unique (Theorem 4.1.14) [pages 165-168 of Sec. 4.1]
  Limit theorems: limits of (s_n+t_n), (ks_n), (k+s_n), (s_nt_n), (s_n/t_n) (Theorem 4.2.1) [pages 171-172 of Sec. 4.2]
  Reading assignment (mandatory): Prove Theorem 4.2.1(a,b,d) (first try to do the proofs yourself, and look in the book only if you get stuck); read Example 4.2.2, and do Practice 4.2.3 [pages 171-173 of Sec. 4.2]
• Lecture 16 (Thu, Oct 9): Limit theorems (cont.): examples of using Theorem 4.2.1 to find limits; if s_n→s, t_n→t, and s_n≤t_n ∀n∈N, then s≤t (Theorem 4.2.4); warning: s_n<t_n does not imply that s<t!; "ratio test": if s_n>0 and lim(s_n+1/s_n)=L<1, then lim(s_n)=0 (Theorem 4.2.7); warning: s_n+1/s_n<1 is not enough to guarantee that lim(s_n)=0 - think of the sequence s_n=n/(n−1); infinite limits; if s_n≤t_n, then s_n→+∞ implies that t_n→+∞, and t_n→−∞ implies that s_n→−∞ (Theorem 4.2.12); if s_n>0, then s_n→+∞ iff s_n→0 (Theorem 4.2.13); examples [pages 172-176 of Sec. 4.2]
  Monotone sequences and Cauchy sequences: increasing, decreasing, and monotone sequences; examples; Monotone Convergence Theorem (Theorem 4.3.3); example - Newton method for finding x^1/2 through the recursively defined sequence s₁=k (where k is any positive number), s_n+1=(s_n+x/s_n)/2 [pages 179-181 of Section 4.3, including Examples 4.3.4-5 and Practice 4.3.6-7]
• Lecture 17 (Tue, Oct 14) Monotone sequences and Cauchy sequences (cont.): more on the Newton method for finding a zero of a nonlinear equation: the method converges quadratically, i.e., the error after the (n+1)st step is approximately equal to the square of the error after the nth step, a numerical illustration using Mathematica (which you can, and should, download for free from the OU IT store); an unbounded increasing sequence diverges to +∞, and an unbounded decreasing sequence diverges to −∞ (Theorem 4.3.8); Cauchy sequences; every convergent sequence is Cauchy (Theorem 4.3.10); every Cauchy sequence is bounded (Theorem 4.3.11); Cauchy Convergence Criterion: a sequence of real numbers is convergent iff it is Cauchy (Theorem 4.3.12) [pages 181-182 of Sec. 4.3]
• Lecture 18 (Thu, Oct 16) Monotone sequences and Cauchy sequences (cont.): a proof of Cauchy Convergence Criterion; remarks on the connection of the Cauchy Convergence Criterion with the Completeness Axiom and the Bolzano-Weierstrass Theorem; a digression: complete ordered fields and complete linear spaces (in which each Cauchy sequence converges); an example of using Cauchy Convergence Criterion: proving that the harmonic series converges by showing that the sequence of its partial sums is not Cauchy and, hence, not convergent [pages 182-184 of Sec. 4.3]
  Subsequences: subsequences - definition and examples; Theorem 4.4.4 (all subsequences of a convergent sequence converge to the same limit); practical uses of this theorem: finding the limit of a convergent sequence by finding the limit of some subsequence (Example 4.4.5), proving that a sequence does not converge by finding two subsequences that converge to different limits (Example 4.4.6); Bolzano-Weierstrass Theorem for sequences: every bounded sequence has a convergent subsequence (Theorem 4.4.7) [pages 186-188 of Section 4.4]
  Reading assignment (mandatory): Read the proof of the Bolzano-Weierstrass Theorem for sequences (Theorem 4.4.7) and think about its conneciton with the Bolzano-Weierstrass Theorem on existence of an accumulation point of a bounded subset of R that contains infinitely many points (Theorem 3.5.6) [page 188 of Sec. 4.4, page 147 of Sec. 3.5]
• Lecture 19 (Tue, Oct 21): Exam 2 [on the material from Sections 3.1-3.5, 4.1-4.3 covered in Lectures 9, 11-16; Cauchy sequences will not be on the exam]
• Lecture 20 (Thu, Oct 23): Subsequences (cont.): every unbounded sequence contains a monotone subsequence that has either +∞ or −∞ as a limit (Theorem 4.4.8) [page 189 of Sec. 4.4]
  Limits of functions: definition of a limit of a function; an equivalent definition of a limit of a function in terms of (open) neighborhoods (Theorem 5.1.2); examples (read Examples 5.1.3, 5.1.5 and 5.1.6 and Practice 5.1.4 and 5.1.7); a detailed proof that limƒ(x)=25 as x→5 for ƒ(x)=x² [pages 196-198 of Sec. 5.1]
• Lecture 21 (Tue, Oct 28): Limits of functions (cont.): sequential criterion for limits (Theorems 5.1.8 and 5.1.10, Corollary 5.1.9), example of usage: limit of sin(1/x) as x→0 doesn't exist; sum, product, multiple, and quotient of two functions; limits of sums, products, multiples, and quotients of two functions (Theorem 5.1.3), examples; one-sided limits [pages 198-202 of Sec. 5.1]
  Continuous functions: definition of a continuous function [pages 205, 206 of Sec. 5.2]
• Lecture 22 (Thu, Oct 30): Continuous functions: equivalent conditions for continuity (Theorem 5.5.2); example: continuity of polynomials and rational functions (wherever the denominator is not zero); proof of the continuity of ƒ(x)=xsin(1/x) for x≠0 and ƒ(0)=0 at x=0 (directly from the definition); test for discontinuity (Theorem 5.2.6); discontinuity of the Dirichlet function at each point of R (Example 5.2.8); the modified Dirichlet function is discontinuous at each c∈Q and continuous at each c∉Q (Example 5.2.9); continuity of sum, product, and ratio of two continuous functions (wherever the denominator is non-zero) (Theorem 5.2.10); continuity of the composition of two continuous functions (Theorem 5.2.12), example of usage (Example 5.2.13) [pages 206-211 of Sec. 5.2]
  A digression: Riemann rearrangement theorem.
  A recommended book: B. Gelbaum, J. Olmsted, Counterexamples in Analysis, Dover, 2003.
  Reading assignment (mandatory): Theorem 5.2.14 (with proof), Corollary 5.2.15, and Example 5.2.16 [pages 211-212 of Sec. 5.2]
• Lecture 23 (Tue, Nov 4): A digression: discussion of Exercise 4.3.15 from Homework 8.
  Properties of continuous functions: bounded functions; the image of a bounded set under a continuous function may be unbounded (consider, e.g., the function tan:(0,π/2)→R); compactness of the image of a compact set under a continuous functions (Theorem 5.3.2); continuous functions on a compact set reach their minimum and maxinum values (Corollary 5.3.3); examples and counterexamples; if a continuous function ƒ:[a,b]→R satisfies ƒ(a)<0<ƒ(b), then ∃c∈(a,b) such that ƒ(c)=0 (Lemma 5.3.5) [pages 215-217 of Sec. 5.3]
  Reading assignment (mandatory): read the proof of Corollary 5.3.3 and the second part of the proof of Lemma 5.3.5 (considering the possibility ƒ(c)>0) [pages 216, 217 of Sec. 5.3]
• Lecture 24 (Thu, Nov 6): Properties of continuous functions (cont.): Intermediate Value Theorem (IVT, Theorem 5.3.6); examples of application: (1) a proof that any polynomial of odd degree has at least one real root (Exercise 5.3/6), (2) a proof that if ƒ:[a,b]→R and g:[a,b]→R are continuous functions that satisfy ƒ(a)≤g(a) and ƒ(b)≥g(b), then ƒ(c)=g(c) for some c∈(a,b) (Exercise 5.3/8), (3) a proof that every closed and bounded subset of the plane can be circumscribed in a square; (4) proving the existence of a zero of a continuous function (useful for numerical zero-finding) [pages 217-219 of Sec. 5.3]
  The derivative: definition of the derivative of a function; examples; a sequential criterion for differentiability (Theorem 6.1.3); differentiability implies continuity (Theorem 6.1.6); derivatives of kƒ, ƒ+g, ƒg, ƒ/g (Theorem 6.1.7); proof that (xⁿ)'=nxⁿ⁻¹ by applying induction and Theorem 6.1.7 [pages 237-242 of Sec. 6.1]
  Reading assignment (mandatory): Example 6.1.8 and Practice 6.1.9 [pages 242, 243 of Sec. 6.1]
  Reading assignment (optional): Theorem 5.3.10 [page 219 of Sec. 5.3]
• Lecture 25 (Tue, Nov 11): The derivative (cont.): Chain Rule (Theorem 6.1.10); example of application of the Chain Rule: computing the derivative of ƒ:R→R defined by ƒ:(x)=x²sin(1/x) for x≠0 and ƒ:(0)=0 for x0 and proving that ƒ':(x) is discontinuous at 0 (Exercise 6.1/6); Weierstrass function - an example of a function that it is continuous everywhere but differentiable nowhere [pages 243, 246 of Sec. 6.1]
  The Mean Value Theorem: if ƒ is differentiable on an open interval (a,b) and assumes its maximum or minimum at c∈(a,b), then ƒ'(c)=0 (Theorem 6.2.1, Fermat's Lemma); Rolle's Theorem (Theorem 6.2.2); Mean Value Theorem (Theorem 6.2.3); if ƒ is continuous on [a,b] and differentiable on (a,b) and ƒ'(x)=0 ∀x∈(a,b), then ƒ is constant on [a,b] (Theorem 6.2.6) [pages 248-251 of Sec. 6.2]
  Reading assignment (mandatory): the proof of the Chain Rule (Theorem 6.1.10); Example 6.1.11 (a continuous function exists everywhere except at 0; Example 6.2.4 and Practice 6.2.5 (applications of the Mean Value Theorem) [pages 243-244 of Sec. 6.1, page 251 of Sec. 6.2]
• Lecture 26 (Thu, Nov 13): The Mean Value Theorem (cont.): if ƒ and g are continuous on [a,b] and differentiable on (a,b) and ƒ'(x)=g'(x) ∀x∈(a,b), then there exists a contstant C such that ƒ(x)=g(x)+C ∀x∈(a,b) (Theorem 6.2.8); strictly increasing and strictly decreasing functions; if ƒ is continuous on [a,b] and differentiable on (a,b) and ƒ'(x)>0 (resp. ƒ'(x)<0) ∀x∈(a,b), then ƒ is strictly increasing (resp. strictly decreasing) on [a,b] (Theorem 6.2.8); if ƒ is differentiable on [a,b] and k is an arbitrary number between ƒ'(a) and ƒ'(b), then ∃c∈(a,b) such that ƒ'(c)=0 (Intermediate Value Theorem for derivatives, Theorem 6.2.9, the proof if optional); Inverse Function Theorem, explicit expression for the derivative of the inverse function [pages 252-254 of Sec. 6.2]
  L'Hospital's Rule: Cauchy Mean Value Theorem (Theorem 6.3.1) [pages 258-259 of Sec. 6.3]
• Lecture 27 (Tue, Nov 18): L'Hospital's Rule (cont.): L'Hospital's Rule (Theorem 6.3.2); limits at infinity (Definitions 6.3.6 and 6.3.7, think about the definition of limƒ(x)=+∞ as x→c); L'Hospital's Rule for the case when the numerator and the denominator tend to infinity (Theorem 6.3.8, skip the proof)); applications: resolving indeterminate expressions of the form 0/0, ∞/∞, 0⋅∞ [pages 259-263 of Sec. 6.3]
  Taylor's Theorem: Taylor's Theorem (Theorem 6.4.2, beginning of the proof) [pages 266-268 of Sec. 6.4]
• Lecture 28 (Thu, Nov 20) Taylor's Theorem (cont.): Taylor's Theorem (Theorem 6.4.2, end of the proof); error bounds based on the formula for the remainder R_n(x) in the Taylor expansion representation ƒ(x)=T_n(x)+R_n(x); the Taylor polynomial T_n(x) as the best polynomial approximation to the function ƒ(x) at a point; binomial formula - Taylor expansion of (1+x)^α for α∉{0,1,2,3,...}; remarks on the interval of convergence of a power series Σa_n(x−x₀)ⁿ [pages 270-272 of Sec. 6.4]
  Metric spaces: definition of a metric and a metric space; examples of metric spaces: (1) R with d(x,y)=|x−y|, (2) a set X with the discrete metric; (3) Rⁿ with the Euclidean metric d₂(x,y)=[Σ_j|x_j−y_j|²]^1/2, (4) Rⁿ with the metric d₁(x,y)=Σ_j|x_j−y_j|, (5) Rⁿ with the metric d_∞(x,y)=max_1≤j≤n|x_j−y_j|, (6) Rⁿ with the metric d_p(x,y)=[Σ_j|x_j−y_j|^p]^1/p; a neighborhood in a metric space [pages 151-154 of Sec. 3.6]
  Reading assignment (mandatory): Example 3.6.4 (neighborhoods in metric spaces); think how the definitions of interior points, boundary points, accumulation points, boundary of a set, open sets and closed sets, interior and closure of a set that were defined for R in Section 3.4 would be modified for a metric space [pages 135, 137, 138 of Sec. 3.4]
• Lecture 29 (Tue, Nov 25): Exam 3 [on the material from Sections 4.3, 4.4, 5.1-5.3, 6.1, 6.2 covered in Lectures 17, 18, 20-26; from Sec. 4.3 only the material on Cauchy sequences may be on the exam (starting with Def. 4.3.9)]

Homework: Homework will be assigned regularly and will be posted on the this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. The lowest homework grade will be dropped. Giving just an answer to a problem is not worthy any credit - you have to write a complete solution which gives your step-by-step reasoning and is written in grammatically correct English. Although good exposition takes time and effort, writing your thoughts carefully will greatly increase your understanding and retention of the material. Always write the proofs on scratch paper first!
You are allowed to discuss the homework problems with the other students in the class. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!
Your homework should have your name clearly written on it, and should be stapled. Please write the problems in the same order in which they are given in the assignment. No late homework will be accepted!

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.

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterm are September 16 (Tuesday), October 16 (Thursday), and November 18 (Tuesday).
The final will be given from 8:00 to 10:00 a.m. on December 9 (Tuesday).
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

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

Policy on W/I Grades : You can withdraw from the course with an automatic "W" from September 2 to October 14 for undergraduate students, or from September 2 to September 26 for graduate students. The period for withdrawing with a grade of W or F is October 27 to December 5 for undergraduate students (and withdrawal requires a petition to the College Dean), or September 29 to December 5 for graduate students. (Petitions to the Dean 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.) Please check the dates in the academic calendar!

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 details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

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.

Good to know:

• The greek_alphabet.
• Some useful notations.