לפוסט זה מצורף אוסף בעיות מספר 2. אם עולות שאלות בקשר לבעיות לפני מפגש התרגול הבא, אפשר לשאול כאן.

Let be a function and a point.

Theorem.

     (1)

if and only if

    (2)

Proof.
1. direction:

by (1).

by definition of the limit of sequence .

Therefore .

2. direction: proof by contradiction.

Assume that the claim is wrong.

Then such that the claim is wrong.

Then the claim is wrong.

Then       (*).

Let be a sequence of positive numbers and a sequence of points constructed as follows:

• ;
• is obtained from (*) for ;
• for all .

Then converges to , and .

Therefore the claim is wrong in contradiction with (2).

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פתרון מילולי

Limits of functions and topology of the real line

• Infinity as the value of the limit: .
• Functions of real variable: the domain, range etc (recall). Examples: polynomial and rational functions, . Compositions: .
• Limit of a function: one-sided, two-sided. Continuity points.
• Sequential limit vs. “standard” limit: equivalence theorem.
• Open and closed subsets on the real line. סביבות וסגור
• Images and preimages. Some algebra of sets: , the same with . Warning: . Operations on infinitely many sets (unions, intersections).
• Local and non-local properties of functions.
• Continuity via open/closed sets.
• Compactness and its implications.

לכניסה זו מצורף אוסף הבעיות הראשון. לחלק מהבעיות המאתגרות יותר מצורפות הדרכות – כמובן, יש יותר מדרך אחת לפתור כל בעיה, וכיוונים מקוריים יתקבלו בברכה. אנא קראו את הבעיות בעיון ונסו לפתור לפחות את חלקן.

Existence of limits and completeness of the real numbers system

• Monotonicity and its implications.
• Nested intervals and their common point
• Boundedness as another property stable by finite alterations
• Converging subsequense of  a bounded sequence
• But why we are so sure that there are no gaps on the real line? And what is a real line?

Construction of the number system: from natural numbers toward scary numbers

• Completion by algebraic operations: from to via . Everything you need to solve linear equations
• Problems  with quadratic equations: irrationalities and negative discriminants. An idea of algebraic number.
• Problems with transition to limit: the ubiquitous and much, much more
• Infinite decimal fractions: completion by “adding limits of monotone sequences”.
• Operations with real numbers: ordered field. Completeness “axiom”.

Limit: the first encounter with infinity

• Introduction and logistics
• Goals of the course
• Infinity: the name of the game in Analysis
• Some examples and counterexamples
  1. Sets and their subsets that have “the same number” of elements
  2. Summation of infinitely many terms: success and failure
  3. Limits in geometry: how to measure the lengths?
  4. Pathological curves on the plane
• Numerical sequences and their limits: the first real encounter with infinity.
  1. “Almost all” vs. “infinitely many“
  2. Intervals and operations on them
  3. Geometric definition of the sequence limit
  4. “Instrumental” definition of the limit
  5. “Standard” definition:
  6. First theorems about limits (limits and arithmetic operations, limit and rearrangements, limit and boundedness, limits and monotonicity)
• Step back: number systems. Integer, rational and real numbers. Sealing the gaps. Completeness

This post marks beginning of the new “blogged” (blog-accompanied) course “Analysis for High School Teachers” within the framework of the Caesarea-Rothschild Program. I will place here brief content of the forthcoming lectures and some relevant material. It may come in a variety of electronic formats, of which most popular are pdf and dejavu. e-Readers for these formats are freely available from the internet.

Blogging rules:

For those not familiar with the blogging subculture (are there still such people?). You are welcome to leave your comments/questions/remarks next to the relevant posts. Please introduce yourself when commenting. My wet dream is having mathematical discussions between the students attending the course on these pages. Don’t be afraid to express yourself and teach others. I promise not to abuse my rights as a moderator here. This venue for interaction is especially important since it is convenient for students which stay away from the teachers for most of the time.

Any language is accepted, though I strongly urge to write in the l.c.d. (= Simple English).

This platform (WordPress) allows for easy insertion of LaTeX code, which makes mathematical discussions here especially pleasant and easy to maintain.

The course will be accompanied by guided seminars led by Gal Binyamini: these seminars will be devoted to discussion of problems and their solutions as well as complimentary material to the main lectures. Gal will also post to this blog.

One of the sources for the course will be the excellent book which equally fascinates both professional mathematicians and high school children. It is in English and you can download an (illegally scanned) copy strictly for your personal use here (21 Mb in pdf format: beware!). I will also try to upload separate relevant sections of the book next to posts on specific lectures.

Finally the two texts concerned with solution of the Infinitesimal Hilbert problem, are put into the polished form (including the publisher’s LaTeX style files). The new revisions, already uploaded to ArXiv, differ from the initial submissions only by corrected typos, a few rearrangements aimed at improving the readability of the texts, and a couple of more references added. There is absolutely no need to read the new revision if you already have read the first one.

Mostly for the reasons of “internal convenience” the complete references are repoduced here:

• G. Binyamini and S. Yakovenko, Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems, posted as arXiv:0808.2950v2 [math.DS], 36 p.p., submitted to Ann. Inst. Fourier (Dec. 2008), accepted (February, 2009)
• G. Binyamini, D. Novikov and S. Yakovenko, On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem, posted as arXiv:0808.2952v2 [math.DS], 57 p.p., submitted to Inventiones Mathematicae (Nov. 2008), accepted (Oct. 2009).

A couple of weeks ago some two-thirds of the conspirators coworkers attended the workshop Equations aux dérivées partielles et théorie de Galois différentielle dit Malgrangefest in Luminy and delivered a talk on their work.

Slides from this talk are now available (static pdf, 2 Mb) for everybody to see.

Best wishes for the New 5769 (תשס”ט) Year to all readers of this blog!

שנה טובה