Introductory Real Analysis I Assignments

MTH 311 (611): Introductory Real Analysis I
Dr. Elmer K. Hayashi
Fall 2003
Assignments

Textbook: Stephen Abbott, Understanding Analysis

Wed, 08/27/2003. Logic and Proof.: Review the rules of logic paying particular attention to quantifiers, negation, proof techniques including proof by contradiction.
Read Sections 1.1-1.2.
Do exercises 1.2.3(a), 1.2.6(c,d), 1.2.7(b), 1.2.8(a-d) to turn in next Monday.

Fri, 08/29/2003. Axiom of Completeness.: At the end of section 1.2, review the Triangle Inequality and a technique for showing two numbers are equal. Begin reading section 1.3 on the axiom of completeness and sups and infs.
Do exercises 1.2.5, 1.3.2, 1.3.6, and 1.3.9 to turn in next Wednesday.

Mon, 09/01/2003. Sup, Inf, Nested Intervals.: Finish section 1.3, and begin section 1.4.
Do exercises 1.3.4, 1.3.5(a), 1.3.8 to turn in next Friday.

Wed, 09/03/2003. Archimedean Property, Density.: Study section 1.4.
Do exercises 1.4.3, 1.4.4, 1.4.5 to turn in next Monday.

Fri, 09/05/2003. Countable and Uncountable Sets.: Finish section 1.4.
Do exercise 1.4.12(a) to turn in next Wednesday.
Think about how to do exercise 1.4.12(b), but you don't need to write it up.

Mon, 09/08/2003. Limit of a Sequence.: Read sections 2.1 and 2.2.
Study examples 2.2.5 and 2.2.6.
Do 2.2.1, 2.2.5 and 2.2.7(a) to turn in this Friday.

Wed, 09/10/2003. Algebraic Limit Theorems: Finish section 2.2 and begin 2.3.
Do 2.2.4, 2.3.6, 2.3.7, 2.3.8 to turn in on Monday.

Fri, 09/12/2003. Cauchy Condensation Theorem.: Read sections 2.3 and 2.4.
Do 2.3.2, 2.3.3, 2.4.1 to turn in on Wednesday.

Mon, 09/15/2003. Bolzano Weierstrass Theorem.: Read section 2.5, and begin section 2.6.
Do 2.5.3, 2.6.1, 2.6.2 to turn in Friday.

Wed, 09/17/2003. Cauchy Convergence Criterion for Series.: Finish section 2.6 and begin section 2.7.
Study 2.6.4 and 2.7.1 for discussion.

Fri, 09/19/2003. Cauchy Convergence Criterion.

Mon, 09/22/2003. Review.

Wed, 09/24/2003. First Hour Exam.: Exam covering Chapters 1 and 2.

Fri, 09/26/2003. Open Sets.: Read section 3.1 and begin 3.2.
Do 2.4.6(b,c), 2.7.6(b), 3.2.1 to turn in on Wednesday.
Those scoring below 80 on the first hour exam must also do 2.6.4 and bring your solution to my office next week.

Mon, 09/29/2003. Closed Sets.: Finish section 3.2.
Do 3.2.2, 3.2.7 and 3.2.11 to turn in Friday.
Do 3.2.3 to discuss in my office.

Wed, 10/01/2003. Compact Sets.: Read section 3.3.
Do 3.3.1, 3.3.2, 3.3.4 to turn in on Monday.
Look at 3.3.5 to increase understanding.

Fri, 10/03/2003. Perfect and Connected Sets.: Read section 3.4.
Do 3.4.1, 3.4.2, (3.4.5 or 3.4.6), 3.4.7 to turn in Wednesday.

Mon, 10/06/2003. Functional Limits.: Read section 4.1 and study section 4.2.
Do 4.2.1(a,c), 4.2.3, 4.2.9 to turn in Friday.

Wed, 10/08/2003. Continuous Functions.: Read section 4.3, study example 4.3.8.
Do 4.3.1 and 4.3.2 to turn in Monday.

Fri, 10/10/2003. Continuous Functions on Compact Sets and Uniform Continuity.: Read section 4.4, study example 4.4.4
Do 4.4.1, 4.4.2, 4.4.3 to turn in Wednesday.

Mon,10/13/2003. Intermediate Value Theorem.: Complete section 4.4.
Look at 4.4.13(a) for practice.

Wed, 10/15/2003. Differentiable Functions.

Fri, 10/17/2003. Fall Break.: No class.

Mon, 10/20/2003.

Wed, 10/22/2003. Second Hour Exam.: Exam covering Chapters 3-5.

Fri, 10/24/2003. Sequences of Functions, Uniform Convergence.: Read section 6.1 and 6.2.
Do exercises 6.2.2 and 6.2.3, 6.2.8(a,b) to turn in Wednesday.

Mon, 10/27/2003. Preservation of Continuity.: Finish section 6.2 and begin 6.3.
Do exercises 6.2.7, 6.2.8(c), 6.2.11(c) to turn in Friday.

Wed, 10/29/2003. Uniform Convergence and Derivatives.: Finish section 6.3.
Do exercises 6.3.2(a), 6.3.4(b), 6.3.5 to turn in Monday.

Fri, 10/31/2003. Uniform Convergence of Series.: Read section 6.4.
Do exercises (6.4.3(a) or 6.4.8), 6.4.5, 6.4.7 to turn in Wednesday.

Mon, 11/03/2003. Power Series.: Begin section 6.5.
Do exercises 6.5.4(a) and 6.5.5 to turn in Friday.

Wed, 11/05/2003. Summation by Parts.: Finish section 6.5.
Do 6.3.3, and 6.3.4(c) to turn in Monday.

Fri, 11/07/2003. Abel's Lemma.: Let f(x) be the series defined in exercise 6.4.6. Prove that f is differentiable in [0,b], 0

Mon, 11/10/2003. Abel's Theorem.: Finish up section 6.5.
Do problems 6.5.4(b) and 6.5.6 to turn in Friday.

Wed, 11/12/2003. The Riemann Integral.: Study the example given on page 198 carefully,
then do problems 7.4.3 and 7.4.6 to turn in Monday.

Fri, 11/14/2003. Uniform convergence and integrability.: Read over sections 7.1-7.4 taking special note of Theorem 7.2.9, Exercise 7.2.5, Theorem 7.4.1, Theorem 7.4.2, Definition 7.4.3, and Theorem 7.4.4.

Mon, 11/17/2003. Integrals of a sequence.: Review chapter 6 and chapter 7 with emphasis on chapter 6 for the exam on Friday.

Wed, 11/19/2003. Review.

Fri, 11/21/2003. Third Hour Exam.: Exam covering Chapters 6-7.

Mon, 11/24/2003. Metric Spaces.: Begin reading section 8.2, pages 222-223.
Do exercises 8.22(a), 8.2.4, 8.2.5(b) to turn in on Wednesday, Dec. 3.

Wed, 11/26/2003. Thanksgiving Holiday.: No class.

Created 07/01/2003. Last modified 11/24/2003. Email to ekh@wfu.edu