Real Analysis I
Fall 2011- The Humke Version
- Text: Understanding Real Analysis by Paul Zorn
- TGWSUF: Paul Humke Office: OMH 304 ext 3410
- Hotline: 645-6440
- Office Hours: TTh 10:00-11:00, MWF 2:00-3:00
- Homework: I'll ask that a few problems from each assignment be written up for grading and will collect these weekly. Each will be worth about 10-15 points.
Keep all of your homework in
a loose leaf 3--ring binder, I'll collect your binder periodically Working together
to solve the problems is absolutely fine and is encouraged!!,
but you must write up your own solutions.
- I'll try to reserve half of Friday for "Solution Day'' where I'll ask you to present
solutions to the problems on the board.
- Quizes: There will be several 10-20 point quizes checking on the basics, definitions, and theorems. I'll let you know what's coming.
- Midterms: There will be two announced 100 point midterm exams
- Written Final: The Written Final, celebrated on Tuesday, Dec. 20, 9:00-11:00 a.m, will be comprehensive and is important -- worth 200 points..
- Grading Scale
In general, here is the course grading scale:
- A 93-100
- A-/B+ 92
- B 85-91
- B-/C+ 84
- C 74-83
- C-/D+ 73
- D 63-72
The Ideas We'll Cover
- The Real Number System: rational, irrational, transcendental numbers, suprema, infima, completeness, nested interval property, denseness of the rational and irrational numbers, cardinality, Cantor-Bernstein-Schroeder Theorem, Cantor diagonalization, Cantor's Theorem.
- Sequences and Series: convergence, algebraic and order limit theorems for sequences and series, monotone convergence theorem, Cauchy condensation test, Bolzano Weierstrass Theorem for squences, Cauchy criterion for convergence of sequences and series, standard tests for convergence of series, rearrangements of series.
- Basic Topology of the Real Line: open sets, closed sets, $G_\delta$ sets, $F_\sigma$ sets, compact sets, Heine-Borel Theorem, perfect sets, connected sets, totally disconnected sets, nowhere dense sets, Cantor's set, Baire Category Theorem.
- Functional Limits and Continuity: the functions of Thomae and Dirichlet, topological and sequential definitions for functional limits, continuity, preservation of connected sets, preservation of compact sets, extreme value theorem, uniform continuity, intermediate value theorem, characterization of set of points of continuity.
- The Derivative: the definition and basic properties, derivatives of the elementary functions, the intermediate value property for derivatives, the mean value theorem, continuity versus differentiability.
- Sequences and Series of Functions: pointwise convergence, uniform convergence and convergence in mean (or probability). uniform convergence preserves continuity, an ocean of examples.
- The Riemann Integral: the geometry of anti-derivatives, the definition of the Riemann integral, properties of the integral, the fundamental theorem of calculus.
Definitions and Theorems
I'd like you to keep a complete, dictionary
(ie. an alphabetized list) of the definitions we learn in the course. Check out Math World to look up any definitions not in the book.
This should be word processed and updated weekly. In addition, I'd like you to keep a separate list of all theorems.
For an intro to the
mathematics wordprocessing program called LaTeX, click
Suggestions for Success
Book Homework will be on Moodle Here's the first assignment though.
- Work on this course every day. Sometimes it may seem that
your ``work'' didn't produce any tangible results, just questions and
frustration. Knowing what questions to ask and knowing what hasn't
worked for you, reflect progress. You should be encouraged and seek
help. This is not an easy course to catch up in should you fall behind.
- Rewrite those homework problems that gave you trouble. Your Homework notebook will be very useful
as you prepare for tests and the exam.
- Quiz yourself on the definitions and theorem statement regularly. A stack of Flash cards can be very effective.
- Don't fall in the trap of copying down every single word of a
proof presented in class. Pay attention to the ideas and make sure that you are
following the logic of the proof. You can then reconstruct
the proof for your notes later that same day. Most proofs are carefully
presented in the text, anyway.
- A source of frustration at the beginning of this course may
be the ease with which you see some of your classmates completing their
assignments while you are struggling. Don't be alarmed. There is a
skill to be learned in this course and it can be learned. It can take
some folks a bit longer than others, though.
- September 16 -- Section 1.1 -- Problems 1,2,4,5,6,8,9,10
- Read this Beginning LaTeX doc and install LaTeX on your laptop.