Lectures: MWF 12:00-12:50pm (Section 44779) and 1:00-1:50pm (Section 44775) in SH 174

Discussion: TuTh 1:00-1:50pm in SSTR 103 (Section 44780) and 10:00-10:50am in SST 120 (Section 44776)

Course description: Introduction to real analysis, including convergence of sequence, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs. Chapters 1-3 (except 3.19, 3.20) will be covered.

Prerequisites: Prerequisites: (MATH 2B or AP Calculus BC) and (MATH 2D or MATH H2D) and (MATH 3A or MATH H3A) and MATH 13. AP Calculus BC with a minimum score of 4. MATH 13 with a grade of C or better.

Textbook: K. Ross, Elementary Analysis, second edition.

The course grade will be determined as follows:

• Homework: 10%. One homework with the lowest score will be dropped. Solutions will be collected every Thursday. Late homework will not be accepted. You are welcome and encouraged to form study groups and discuss homework with other students, but you must write your solutions individually.
• Midterm Exam 1: 25%, Wednesday, October 24, in class. Covers everything covered in class up to, and including, October 17.
• Midterm Exam 2: 25%, Monday, November 19, in class. Covers everything covered in class up to, and including, November 9.
• Final Exam: 40%, Wednesday, December 12, 1:30--3:30pm, in ICS 174.

There will be no make-up for the exams for any reason. A missed midterm exam counts as zero points, with the following exception. If you miss a midterm exam due to a documented medical or family emergency, the exam's weight will be added to the weight of the final exam.

• Friday, September 27

  Review: Appendix on Set Notation. Natural numbers. Induction. (Section 1.1).
  Homework 1 (due October 4): 1.1, 1.2, 1.3, 1.6, 1.7, 1.8, 1.9, 1.12, 2.1, 2.2, 2.4, 2.8.

• Monday, October 1

  Integer and rational numbers. Rational Root Test (a.k.a. Rational Zeros Theorem). Section 2 and part of Section 3 were covered.

• Wednesday, October 3

  Order axioms. Consequences of field and order axioms (Theorems 3.1, 3.2). Absolute value (Definition 3.4).
  Homework 2 (due October 11): 3.1, 3.3, 3.4, 3.5(b), 3.6, 3.7(c), 3.8, 4.1, 4.2, 4.3, 4.4.

• Friday, October 5

  Distance. Triangle inequality (Theorem 3.5 iii, Corollary 3.6). Bounded sets. Supremum and infimum (Section 4 through Example 4).

• Monday, October 8

  Completeness axiom. Archimedian property. (Below Definition 4.3 through 4.6).

• Wednesday, October 10

  Floor and ceiling functions (not from book). Denseness of the set or rational numbers (4.7).
  Homework 3 (due October 18): Find the suprema and prove your claims in problems 4.1 b,c,j,l; find the infima and prove your claims in problems 4.1 m,n; solve problems 4.5, 4.6, 4.7, 4.8, 4.11, 4.12, 4.14(a).

• Friday, October 12

  Study Section 1.5 at home (Infinite suprema and infima). Convergence of sequences (Sections 1.7-1.8).

• Monday, October 15

  Proving convergence of sequences (examples similar to ones in Section 1.8 and Theorem 9.7).
  Sample Midterm exams: sample-midterm1a problems 1, 2(a), 4; sample-midterm1b problems 1, 2, 3, 4(a,b).

• Wednesday, October 17

  Convergence implies boundedness (Theorem 9.1). Limit theorem for the sum (Theorem 9.3)
  Homework 4 (due October 25): 7.2, 7.4, 8.2, 8.4, 8.6, 8.7(a), 8.9(b), 8.10, 9.1, 9.2, 9.3, 9.4.

• Friday, October 19

  More limit theorems: for differences, products, and ratios (Section 9).

• Monday, October 22

  Divergence of (sin n) (not from the book). Limit and the order relationship. The Squeeze Theorem (Exercise 8.5).

• Wednesday, October 24

  Midterm Exam 1.
  The exam covers everything that we covered in class up to, and including, October 17.
  Homework 5 (due November 1): 5.1 (use infinity symbols for unbounded sets), 9.6, 9.8, 9.9, 9.10, 9.11, 9.13, 9.12 (b) (a version of ratio test for infinite limits), 9.15, 9.16 (a,b).

• Friday, October 26

  Infinite infima, suprema (Section 1.5) and limits (Section 2.9). Ratio test (Exercise 9.12a).

• Monday, October 29

  Examples for ratio test (9.14). Monotone sequences (3.3). Monotone convergence theorem.

• Wednesday, November 1

  Applications of the monotone convergence theorem: decimal representations of real numbers, limits of recursively defined sequences, the number e (3.3.6).
  Homework 6 (due November 8): 9.17, 9.18, 10.1, 10.2, 10.4, 10.7, 10.8, 10.9, 10.10, 10.11, 10.12.

• Friday, November 2

  Nested intervals theorem. Subsequences: definition and examples (Section 2.11).

• Monday, November 5

  Bolzano-Weierstrass theorem (11.5). Cauchy sequences (10.8-10.11).

• Wednesday, November 7

  An application of Cauchy criterion: Banach fixed point theorem.
  Homework 7 (due November 15): 11.1 (don't specify the function sigma); 11.2, 11.4, 11.7 (recall from Math 13 that the set of rational numbers is countable), 11.11; prove that a sequence of positive numbers is unbounded if and only if there is a subsequence that diverges to +infinity; problems 30, 35, 41 from this problem set.

• Friday, November 9

  Lim sup and lim inf (parts of 2.10, 2.11, 2.12). Study Theorem 10.7 at home.
  Sample Midterm exams: sample-midterm1b problems 1 (for lim inf, lim sup), 4c, 5, 6; sample-midterm2a problems 2, 3, 4, 5, 6; sample-final1 problems 1a, 3.

• Wednesday, November 14

  Series: definition, n-th term criterion, Cauchy criterion (14.1-14.4).
  Homework 8 (due November 21 - WEDNESDAY!): 12.4, 12.6, 12.10, 14.1(a,c,d,e,f), 14.2, 14.3, 14.4(b,c), 14.6, 14.7, 14.8.

• Friday, November 16

  Series: comparison test (14.6) and the limit comparison test.

• Monday, November 19

  Midterm Exam 2.

• Wednesday, November 21

  Series: absolute convergence (14.7), root test (14.9), ratio test (14.8), integral test (15.2).
  Homework 8 (due November 29): study the Alternating Series Theorem (15.3) and Theorem 15.1 (don't turn them in); do problems 14.9, 14.4(a), 14.11, 14.12, 14.14, 15.1, 15.2, 15.4, 15.6.

• Monday, November 26

  Continuous functions: sequential and epsilon-delta definitions. Examples. (17).

• Wednesday, November 28

  Operations on continuous functions (17.3, 17.4, 17.5). Extreme value theorem (18.1).
  Homework 9 (due December 6): 17.2, 17.8, 17.9(a,b), 17.10(b,c), 17.12, 17.13, 17.14, 18.4, 18.6, 18.10.

• Friday, November 30

  Intermediate value theorem (18.2). Applications: solving equations numerically, fixed point theorems.

• Monday, December 3

  Inverse function theorem (18.4). Metric spaces: definition (13).

• Wednesday, December 5

  Metric spaces: examples, convergence and completeness. Completeness of R^k (13.2-13.4).
  Sample final exams: sample-midterm1a problems 2(b), 3, 4, 5, 6; sample-final1 problems 2, 4(a,b), 5; sample-final2 problems 1, 4, 5, 7(b); sample-final3 problems 1, 2, 5 (without C), 7, 8, 9 (without differentiability), Extra Credit.

• Friday, December 7

  Review.