# Math 403 - Fall 2011               Classical Analysis I

Schedule of classes :
Section 1, M W F 1:25 PM - 2:15 PM, in 014 Henderson Bldg (Schedule n. 641881)
Section 2, M W F 12:20 PM - 1:10 PM, in 116 Osmond Lab (Schedule n. 641884)

Office Hours (subject to change): FRIDAY 9:30 - 11 AM, WEDNESDAY 5 - 6:30 PM.
EXTRA OFFICE HOURS BEFORE FINAL EXAM: MONDAY DEC 12, 11 AM - 12 PM.

Final Exam : MONDAY DECEMBER 12, 2:30 - 4:20 PM IN 112 WALKER.

Midterm Exams: WEDNESDAY SEPTEMBER 21, 6:30 -7:45 PM in 101 OSMOND
TUESDAY OCTOBER 25, 6:30 -7:45 PM in 201 THOMAS

FINAL EXAM is comprehensive (List of topics). Review in class Friday Dec 9.
Final Review Problems.

MIDTERM 1 will cover Sections 1.1 - 2.1 included (List of topics). Review in class Monday Sep 19. EXTRA OFFICE HOUR on Monday Sep 19, 5:30 to 6:30 PM.
Midterm 1 Review Problems.

MIDTERM 2 will cover Sections 2.3-2.4 and 3.1-3.4 included (List of topics). Review in class Monday Oct 24. OFFICE HOURS on Monday Oct 24, 5:00 to 6:30 PM. NO OFFICE HOURS WEDNESDAY Oct 26 (due to travel).
Midterm 2 Review Problems.

Syllabus.

Course Topics: from Mathematical Analysis by Andrew Browder, Springer.
1. Properties of real numbers (chapter 1): infinite sets and countability (1.3), ordered fields and completeness (1.5), the field of real numbers (1.7 and 1.9). The other sections in Chapter 1 are to be reviewed by the students and will not be covered in class.
2. Sequences and Series (Chapter 2): sequences: examples, properties and monotone series (2.1), series: examples and convergence criteria (2.3), rearrangements and operations with series (2.4). We will skip continued fractions (2.2) and unordered series (2.5).
3. Continuous functions on intervals (Chapter 3): limits and continuity (3.1), Maxima and minima on closed intervals and the Intermidiate Value Theorem (3.2), uniform continuity (3.3), and sequences of functions (3.4). We will review the exponential and trigonometric functions (3.5 and 3.6), skipping some proofs.
4. Derivatives (Chapter 4): definition and elementary properties (4.1), convex functions (4.3), derivative of inverse and composite functions (4.4), l' Hopital's Rule (4.5), higher order derivatives (4.6), analytic functions (4.7). You will be asked to review derivatives of elementary functions (4.2).
5. The Riemann integral (Chapter 5): Riemann sums (5.1), examples and conditions for the existence of the integral (5.3), integrals of sequences and series (5.5). We will skip improper integrals (5.6). You will be asked to review properties of the integral (5.3) and the Fundamental Theorem of Calculus (5.4), which are convered in Calculus courses.
6. Topological spaces (Chapter 6): definitions and basic examples (6.1), continuous maps (6.2), metric spaces (6.3). If time allows, we will briefly discuss compact sets (6.6) and connected sets (6.7).

Homework Problems: solutions will be available on ANGEL ( under the "lessons" tab) roughly one week after the problems are posted.
1. Chapter 1, pages 23-25: 2, 4, 14, 18 ( NOT COLLECTED OR GRADED, this homework is for your practice only.)
2. Chapter 2, pages 50-51: 1--6. DUE MONDAY SEPTEMBER 12.
3. Work out the following problems: 1) Show that a convergent sequence is bounded. 2) Prove Proposition 2.16, a), b), c). Give an example showing that you can have strict inequality in part c). 3) Prove Proposition 2.16, Part e). You may assume part d) (proved in class). What can you conclude about subsequences of a convergent sequence? Justify your answer carefully. DUE MONDAY SEP 19.
4. Solve the following problems: 1) Use the Bolzano-Weierstrass theorem to show that a Cauchy sequence converges. 2) Let A be a (non-empty) bounded subset of the real line, prove that there exists a sequence, the terms of which are all elements in A, that converges to Sup(A). 3) Prove that a sequence converges if and only if all its subsequences converge to the same limit. DUE MONDAY SEPTEMBER 26.
5. Chapter 2, pages 51-52: 12 (Hint: it may be useful to review summation by parts formula to find the sum of the series), 13, 14 ( Hint: reduce to the case where the limits are real numbers. Using the definition of upper (or lower) limit, find a formula relating a_n to a_N for n>N, N large enough), 16. DUE MONDAY OCTOBER 3.
6. View this file for the homework. DUE MONDAY OCTOBER 10.
7. Chapter 3, pages 70-71: 4, 5 c), 6 a) and c), 7. DUE MONDAY OCTOBER 17.
8. Chapter 3, pages 71-72: 9 -- 12. DUE MONDAY OCTOBER 24.
9. Chapter 3, page 73: 15, 16, 18, 19. DUE MONDAY OCTOBER 31. Chapter 4, pages 93-94: 1, 3 (you may use without proof that f is constant if f'=0), 7 (Hint: use the Mean Value Theorem, Theorem 4.22, you may also use without proof that g is increasing if its derivative g' is non-negative), 8 (Hint: use Ex. 7 to deduce that the exponential function is convex). DUE MONDAY NOVEMBER 7.
10. Chapter 4, pages 93-94: 6 (Hint: use the MVT), 10 (Hint: rewrite the limit in order to apply L'Hospital's rule), 11 a), b), c), 12. DUE MONDAY NOVEMBER 14.
11. Chapter 4, pages 95: 15, 16 (you may assume that M_0,M_1, and M_2 are all numbers, so in particular the function is bounded), 18 (Hint: do a proof by contradiction. Let then X be a zero that is not an isolated point, and expand f in power series centered at Z. Then, observe that if f is not identically zero, there is at least one nonzero coefficient. Conclude that f must have a certain form near Z). Chapter 5, page 118: 3 (Hint: look at Definition 5.7 and Theorem 5.10). DUE MONDAY NOVEMBER 28.
12. Chapter 5, pages 119-120: 7, 12 (follow the hint of integrating by parts repeatedly), 13 (please NOTE the formula in the book is incorrect, please use the formula given on monday in class), 16 (Hint: first use a) to write the integral - f(0) as one integral. Then split the interval [-1,1] into two parts, one where the integral is small for large n because of hypothesis b), the second where the integral is small because of the continuity of f at zero). DUE MONDAY DECEMBER 5.
13. Chapter 6, pages 150-151: 1, 5, 7, 9. NOT COLLECTED OR GRADED.