Math 450 – An Introduction to Real Analysis – 3 credits

Professor:  M. Greeley

Office:  Science Center 3-170

Telephone: 617 287 6456

Office Hours:  Tu/Thu 11-12 or by appointment

Course description:  A rigorous treatment of the calculus of functions of one real variable.  Emphasis is on proofs.  Includes discussion of topology of real line, limits, continuity, differentiation, integration and series.

Prerequisite:  Math 310 or Math 354

Textbook:   Analysis with an Introduction to Proof, by Steven R. Lay

Homework: Due the following week on Thursday. Homework must look “professional”; i.e. no scratch-work, use rulers or graphing paper where appropriate.

Grade:  Homework 20%, Midterm 40%, Final Exam 40%

Section                        Topic

10                    Natural Numbers and Induction

11                    Ordered Fields

12                    The Completeness Axiom

13                    Topology of the Reals

14                    Compact Sets

16                    Convergence

17                    Limit Theorems

18                    Monotone Sequences and Cauchy Sequences

19                    Subsequences

20                    Limits of Functions

21                    Continuous Functions

22                    Properties of Continuous Functions

23                    Uniform Continuity

25                    The Derivative

26                    The Mean Value Theorem

27                    L’Hopital’s Rule

28                    Taylor’s Theorem

29                    The Riemann Integral

30                    Properties of the Riemann Integral

31                    The Fundamental Theorem of Calculus

32                    Convergence of Infinite Series

33                    Convergence Tests

34                    Power Series

35                    Pointwise and Uniform Convergence

36                    Applications of Uniform Convergence

37                    Uniform Convergence of Power Series

Homework Assignments:

Due                  Page

2/4                   91: 4*, 8*, (13c)*, 16*

2/11                 102: 1,2,(3cj)*, 4*, 10*,( Why can’t Theorem 12.7 be applied be applied to 12.2a?)*, (12.2ab)*

2/18                 112: 1, 2*, 6*, 10*, 12*

2/26                 120: 1, 3, 4*, 5, 6*, 8*, 16*

127: 2*, 3, 4*, 8*, 12* (Hint on 12: 1st, show it’s closed by showing that  and 2d show that it’s bounded (if not, then the infinite

subset property will be violated)

3/4                   146: 1, 2*, 4*, 6*, 10*

(1)*  (use the definition of convergence)

(2) 10.14* (Hint: use induction)

3/11                 154: 2*, 6*, 9*, 16*

3/25                 161: 1, (2ab)*, 4*, 6*, 10*

168: 1, 2*, 8*

4/1                   177: 2*, 4*, 5c, 6*, 9* (in your own words!)

4/8                   186: 2*, 6*, 10*, 11* (in your own words), 16*

4/15                 (16.13)*

192: 2*, 4*, 6*, 11*, 12*

199: 2*, (4ab)*, 6*

4/22                 216: 2*, 6*, 10*

224: 2*, 4*, (5acfh)*, 25.6*, 12*

4/29                 249: 2*, 4*, 6*, 11*, 14*

5/11/04            257: 2*, 5, 6*, 8* (hint: f integrable implies f bounded.

also 11*

264: 4*, 6*, 10*, 16*