Math 452/502 Course page Fall 2003

Math 452/502 Fall 2003 (08/25/03 - 12/11/03)
Introduciton to Real Analysis

Course Information:

Instructor:	Shangbing Ai, MDH 320, Phone: (256) 824-2229, Email: ais@email.uah.edu, Webpage: www.math.uah.edu/ai
Class meetings:	TR 5:30-6:50 pm, MDH 329
Office Hours:	TR 3:30-5:00 pm or by appointment
Prerequities:	MA 330 and MA 442, or approval of instructor. This course is taught as MA 452/502.
Textbook:	Analysis with an Introduction to Proof, 3rd Edition, by Steven R. Lay.
Grading:	Homework (30%), two in class tests (20%+20%), the final (30%).
Test Dates:	Test 1 (Thursday, 10/9), Test 2 (Thursday, 11/13), Final (12/11, 6:30 - 9:00 pm).
Course Objectives:	Sequences, limits, continuity, differentiation of functions of one real variable, Riemann integration, uniform convergence, sequences and series of functions, power series, and Taylor series.

Main Topics and Schedule (tentative)

Chapter 3. The Real Numbers

Sec. 10: Natural numbers and induction (well-ordering property of N (Axiom 10.1), Principle of mathematical induction (Theormem 10.2), exmaple 10.3)
Exercises (P.92): 10.4, 10.14
Sec. 11: Ordered fields (11 axioms for fields and 4 order axioms for fields, Theorem 11.7, the absolute value, Theorem 11.9)
Exercises (P. 102): 11.2, 11. 6, 11.7
Sec.12: The Completeness Axiom (supremum, infimum, the completeness axiom, the density of rational numbers and irrational numbers)
Exercises (P. 112): 12.1, 12.2, 12.3, 12.11 (Hand in 12.3, 12.4, 12.11 on Tuesday, Sept. 9)
Sec.13: Topology of the reals (neighborhood, a deleted nigherborhood, interior points, boundary points, closed sets, open sets, Theorems 13.10 and 13.11, accumulation points, isolated points, the closure of a set, Theorem 13. 17)
Exercises (P. 120): 13.1 - 13.6 (Hand in 13.3, 13.4, 13.5 on Thursday, Sept. 11)
Sec. 14: Compact Sets (compact sets, open cover, subcover, Heine-Borel Theroem, Bolzano-Weierstrass Theorem)
Exercises (P. 127): 14.1, 14.2, 14.3, 14.8, 14.9 (Hand in 14.3, 14.8)
Solution to Problem 12.11

Chapter 4. Sequences

Sec. 16: Convergence of Sequences (Defintion of the convergence of sequences by epsilon - N language, boundedness of the convergence sequences, uniqueness of the limit)
Exercises (P. 146): 16.4 (b), (d), (e), 16.5 (c), (e), (f), 16.12 (Hand in all these problems)
Sec. 17: Limit Theorems (limit theorem, ratio test, infinite limits)
Exercises (P. 154): 17.5 (d), (h), (i), (k), 17.13, 17.15 (c) (Hand in all these problems)
Sec. 18: Monotone sequences and Cauchy sequences
Exercises (P. 161): 18.3 (c), (e), 18.4, 18.5, 18.6, 18.11 (Hand in all these problems)
Test 1 (Thursday, Oct. 9. It covers the materials from Chapters 3 and 4)
Sec. 19: Subsequences (subsequences, limit superior and limit inferior)
Exercises (P. 168): 19.3, 19.4, 19.11, 19.14
Test 1 (supremum and infimum, the completeness Axiom of R, interior pts, boundary pts, accumulation pts, accumulation pts, open sets, closed sets, closure, compact sets, open cover and subcover, Heine-Borel theorem, Bolzano-Weierstrass theorem, nested interval theorem, limit definition definition, the uniqueness of the limit, the boundedness of convergence sequence, limit theorem, ratio test, monotone convergence theorem, Cauchy sequence, Cauchy convergence theorem, the test problems are similar to the homework problems)
Solutions to Problems 19.9, 19.11, 19.14

Chapter 5. Limits and Continuity

Sec. 20: limits of Functions
Exercises (pp. 178): 20.3 (f), (h), 20.4 (c), 20.6, 20.14 (Hand in all these problems.)
Sec. 21: Continuous Functions
Exercises (pp. 186): 21.5, 21.6, 21.8, 21.11, 21.16 (Hand in all these problems.)
Sec. 22: Properties of Continuous Functions
Exercises (pp. 194): 22.4, 22.6, 22.7, 22.9, 22.15 (Hand in all these problems.)
Sec. 23: Uniform continuity
Exercises (pp. 199): 23.3 (b), (d), (f), (g) 23.4, 23.5, 23.11 (Hand in all these problems.)

Chapter 7. Integration

Sec. 29: The Riemann Integral
Exercises (pp. 248): 29.7, 29.8, 29.9, 29.11, 29.14 (Hand in all these problems.)
Sec. 30: Perperties of the Riemann Integral
Exercises (pp. 251): 30.5, 30.6, 30.7 (Hand in all these problems.)
Sec. 31: The Fundamental Theorem of Calculus
Exercises (pp. 264): 31.8, 31.10, 31.12, 31.13, 31.18, 31.19

Chapter 6. Differentation