Humboldt State University Fall 2003
Instructor: Walden Freedman
Office: Library 40
Office Hours: MWF
3:00-4:00 pm, TR 1:00-2:30 pm, or by
appointment
Office Phone: 826-4763 Math Dept Phone: 826-3143
Math Dept Fax: 826-3140 E-mail: wf6@humboldt.edu
·
Next to my office is the math
department’s informal library. You can
find a number of additional real analysis texts to consult.
·
I have a large collection of real
analysis books in my office, courtesy of Dr. C. Biles.
You are welcome to take a look.
·
Check out the real analysis links on my
webpage: http://www.humboldt.edu/~wf6/links.htm#Real
Analysis
Midterm I: Monday,
September 29
Midterm II:
Wednesday, November 12
Final Exam: Friday,
December 19, 10:20—12:20. Make your
travel plans accordingly.
(a) Sets and functions: Sets and
elements of sets; Subsets; Equality of sets; Intervals in R; The natural numbers N,
the integers Z, and rational numbers Q.
The empty set; Union, intersection, and complements of sets; Disjoint sets;
Indexed families of sets and set operations on them; Symmetric difference; Ordered pairs;
Cartesian products; Functions; Domain, range, and codomain; Surjective (onto) functions;
Injective (one-to-one) functions; Bijections; Characteristic (indicator) functions; Action
of functions on sets; Image and preimage; Composition of functions; Inverse functions;
Equinumerous sets; Denumerable sets; Countable and uncountable sets; Uncountability of R
(b) Induction: Induction;
Well-ordering property of N
(c) Ordered fields: Fields; Commutative and distributive laws;
Trichotomy, and other order laws; Ordered fields
(d) The real number system and its
topology: Upper and lower bounds; Bounded sets; Maximums and minimums of sets;
Supremum and infimum; Completeness axiom; Archimedean property of R; Denseness of Q
in R; Neighborhoods in R; Deleted neighborhoods; Interior and boundary
points; Open and closed sets; Accumulation points; Isolated points; Closure, boundary, and
interior of a set; Compact sets; Open covers and subcovers of sets; Heine-Borel Theorem;
Bolzano-Weierstrass Theorem
(e) Metric spaces: Triangle
inequality; Euclidean metric; Bounded sets in metric spaces; Dense subsets; Separable
metric spaces
(f) Sequences: Convergence; Limits
of sequences; Divergence; Bounded sequences; Uniqueness of limits; Boundedness of
convergent sequences; Algebraic properties of limits; Infinite limits; Divergence to ±8;
Monotone sequences; Monotone Convergence Theorem; Cauchy sequences; Cauchy Convergence
Criterion; Contractive sequences; Subsequences; Bolzano-Weierstrass Theorem for sequences;
Limsup and Liminf; Unbounded sequences
(g) Infinite series and
power series:
Partial sums; Infinite series; Convergent and divergent series; Harmonic series; Cauchy
criterion for series; Geometric series; Comparison test; Absolute convergence; Conditional
convergence; Ratio and root tests; Integral test; Alternating series test; Rearrangements
of series; Power series; Radius of convergence; Interval of convergence
(h) Limits and continuity: Limits of
functions; e-d proofs; Sequential criterion for limits of functions; Algebraic properties
of limits; Right-hand and left-hand limits; Continuity of a function at a point;
Continuous and discontinuous functions; Algebraic properties of continuous functions; Boundedness of functions; Continuous image of a
compact set is compact; Intermediate Value Theorem; Uniform continuity
(a) Grading: The final course grade will be
determined as follows:
Homework 50%; Two midterm exams 15%
each; Final exam 20%
Warning! I am not a
slave to any grading "formula". For most students, the following rough
guide works well:
88-100% = A,
75-87% = B, 62-74% = C, 49-61% = D, and =48% =
F
When the homework and exam scores are consistent,
grading is usually not a challenge. However, when the homework and exam scores
are inconsistent, I rely on my professional judgment and not on a formula. The
bottom line is: did you learn the material, not how many points you accumulated.
(b) Homework: I will
assign homework at nearly every class meeting. Due
dates will be announced in class. Generally,
homework will be due the day of the next class meeting by 4:00 pm. You may leave homework in the appropriate wall
folder next to my office door if I am not in my office.
Do not leave homework in my mailbox in the math office. I will not accept late homework under any
circumstances; however, there will also be some opportunities to do extra credit and I
will drop the lowest two homework scores. Each
homework assignment will be worth 10 points. I
may also give bonus homework points for excellence in content and presentation. For example, well-done computer-generated
graphics (if appropriate), outstanding solutions, etc.
Due to time constraints, I will not grade all problems assigned, but
solutions will be available for all problems assigned. Be aware that to receive full
credit on homework, you must show a good effort on all problems
assigned. Be advised that homework for this
course will take a good deal of time and effort. I
advise you to begin working on a homework assignment as soon as it is assigned. It is all too easy to fall behind with the
homework, a recipe for disaster in this course: later topics and techniques absolutely
require and refer to those previously covered.
Handwritten homework must be prepared neatly on clean paper, one side
only. You must staple multiple pages together. I will return sloppy, disorganized work to
you unread with a grade of zero. I suggest doing each assigned problem twice. Do the first
draft on scratch paper. When you know exactly
how a solution goes, transcribe it neatly onto a separate sheet. See the “guidelines for homework
assignments” for how to write up your assignments.
(c) E-mail and
course webpage: I will frequently use e-mail to contact
individuals as well as the class as a whole, so please check via Records and
Registration (http://www.humboldt.edu/~records/enrollment_svcs/what_registration.shtml)
that your email address on file is current and usable.
Otherwise, you may miss important information during the semester. On the course
webpage http://www.humboldt.edu/~wf6/m415/
you will find a link to an MS-Excel sheet containing current homework and exam scores for
the entire class. This will enable you to
keep track of your progress, and inform me of any discrepancies. You will also be able to find a copy of this
document (possibly updated), as well as other information.
Documents are in MS-Word, MS-Excel, or possibly PDF format. Please contact me if you need help with these
formats.
Week |
Topics |
Week #1 Aug 25—Aug 29 |
Basic
set operations; indexed families of sets; Functions; Image and preimage; Countable sets;
Induction; Strong induction; Ordered fields |
Week #2 Sept 3—Sept 5 |
Induction; Ordered fields; Upper and lower bounds; Completeness axiom; Supremum and infimum (Sept. 1st is Labor Day, a holiday) |
Week #3 Sept 8—Sept 12 |
Bounded
functions; Archimedean property; Q is dense in R; |
Week #4 Sept 15—Sept 19 |
Accumulation
points; closure of a set; cl S is a closed set; open covers, |
Week #5 Sept 22—Sept 26 |
Compact sets:
Heine Borel Theorem, Bolzano-Weierstrass Theorem, Nested Intervals Theorem, Metric Spaces;
Review |
Week #6 Sept 29—Oct 3 |
Midterm
I; Sequences, and convergence of sequences; uniqueness of limits; boundedness of
convergent sequences; examples |
Week #7 Oct 6—Oct 10 |
More on limits,
characterization of accumulation points in terms of convergent sequences; limit theorems;
Infinite limits; Monotone and Cauchy sequences |
Week #8 Oct 13—Oct 17 |
Monotone
sequences, inductively defined sequences; Cauchy sequences; proof that every Cauchy
sequence is convergent; Subsequences |
Week #9 Oct 20—Oct 24 |
Limsup and
liminf ; Unbounded sequences; Convergence of infinite series |
Week #10 Oct 27—Oct 31 |
Proof of, and
examples of Comparison Test; Limit Comparison Test; Ratio Test; Absolute convergence |
Week #11 Nov 3—Nov 7 |
Proofs of root
and integral tests; Alternating series test; power series |
Week #12 Nov 10—Nov 14 |
Review; Midterm
II; Rearrangements of series |
Week #13 Nov 17—Nov 21 |
Limits of
functions; "epsilon-delta" proofs, Algebraic properties of limits; Right-hand
and left-hand limits; continuous functions |
Week #14 Dec 1—Dec 5 |
Intermediate
Value Theorem; Boundedness of continuous
functions; Continuous
image of a compact set is compact; Uniform continuity |
Week #15 Dec 8—Dec 12 |
Uniform
continuity; Review |
General advice:
·
Use your old Calculus text as a
reference, especially for infinite sequences, series, power series, and continuity. You will find many of the same theorems and proofs
in our course, and the presentation may be easier for you to follow. There are also some interesting problems in the
exercises.
·
Read the textbook before you come to
class—that way when you come to class, you will already be somewhat familiar with the
ideas and we can spend our time more productively.
·
Plan to spend about 15 hours a week
studying real analysis. This means that a
good part of the learning experience in the course will happen outside the classroom. Good study habits and a serious attitude are
important.
·
Form a group of other students with
whom to study. There are fewer better ways to
increase your understanding of a subject than to try to explain it to a classmate.
·
Always read over your lecture notes,
and preferably rewrite them, noting any items you didn’t fully understand, within 24
hours of the lecture.
Office: Library 40 (basement) Webpage: http://www.humboldt.edu/~wf6 E-mail: wf6@humboldt.edu
Department of Mathematics, Humboldt State University, 1 Harpst St,
Arcata, CA 95521 Phone: (707) 826-4763