Sergei Yakovenko’s Weblog

Thursday, November 19, 2009

אוסף בעיות מספר 2

Filed under: Rothschild course, problems & exercises — galbin @ 1:17

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

Wednesday, November 18, 2009

A sample theorem, a sample proof

Filed under: Rothschild course, lecture, problems & exercises — Sergei Yakovenko @ 6:14

Let f\colon X\to\mathbb R be a function and a\in\mathbb R a point.

Theorem.

A=\lim_{x\to a}f(x)      (1)

if and only if

\forall\{x_n\}_{n=1}^\infty\subseteq\mathbb R\smallsetminus\{a\}\ \lim x_n=a\implies \lim f(x_n)=A.     (2)

Proof.
1. (1)\implies(2) direction:

\forall \varepsilon >0\ \exists \delta>0:\forall x\ 0<|x-a|<\delta \implies |f(x)-A|<\varepsilon by (1).

\forall \{x_n\}\in\mathbb R\smallsetminus\{a\}\ \lim x_n=a\implies\exists N:\ \forall n\ge N\ 0<|x_n-a|<\delta by definition of the limit of sequence \{x_n\}.

Therefore \forall \{x_n\}\in\mathbb R\smallsetminus\{a\}\ \lim x_n=a\implies\forall\varepsilon>0\ \exists N:\ \forall n\ge N\ |f(x_n)-A|<\varepsilon.

2. (1) \Longleftarrow (2) direction: proof by contradiction.

Assume that the claim \boxed{\lim_{x\to a}=A} is wrong.

Then \exists \varepsilon_*>0 such that the claim \boxed{\exists\delta>0,\ \forall x\ 0<|x-a|<\delta\Longrightarrow |f(x)-A|<\varepsilon} is wrong.

Then \exists \varepsilon_*>0\ \forall\delta>0 the claim \boxed{\forall x\ 0<|x-a|<\delta\implies |f(x)-A| <\varepsilon} is wrong.

Then \exists \varepsilon_*>0\ \forall\delta>0\ \exists x=x_\delta:\ \bigl\{\ 0<|x-a|<\delta\ \&\ |f(x)-A|\ge \varepsilon\bigr\}       (*).

Let \delta_n>0 be a sequence of positive numbers and \{x_n\} a sequence of points constructed as follows:

  • \delta_1=1;
  • x_n=x_{\delta_n} is obtained from (*) for \delta=\delta_n;
  • \delta_{n+1}=\tfrac12|x_n-a| for all n=1,2,3,\dots.

Then \{x_n\} converges to a, x_n\ne a and \forall n\in\mathbb N\ |f(x_n)-A|\ge\varepsilon_*.

Therefore the claim \boxed{\forall\{x_n\}_{n=1}^\infty\subseteq\mathbb R\smallsetminus\{a\}\ \lim x_n=a\implies \lim f(x_n)=A} is wrong in contradiction with (2). \blacksquare

___________________________________

פתרון מילולי

 

מחד גיסא, נראה שאם הפונקציה f שואפת ל-A ב-a, אז לכל סדרה x_n המתכנסת ל-a, הסדרה (f(x_n שואפת ל-A. אכן, עבור כל קטע פתוח I סביב A, קיימת סביבה נקובה N של a המועתקת ע”י f ל-I. הסדרה x_n נמצאת כמעט כולה ב-N, ולכן התמונות (f(x_n נמצאות כמעט כולן ב-I. לכן, לפי ההגדרה, (f(x_n מתכנסת ל-A, כנדרש.

 

מאידך גיסא, נראה שאם הפונקציה f אינה שואפת ל-A ב-a, אז יש סדרה x_n כך ש-(f(x_n אינה שואפת ל-A. אכן, קיים קטע פתוח I סביב A כך שקיימות נקודות קרובות כרצוננו ל-a (ושונות מ-a) שתמונותיהן ע”י f לא נמצאות ב-I. מתוך הנקודות הללו ניתן לבחור סדרה אינסופית x_n המתכנסת ל-a. אבל אז ברור ש-(f(x_n אינה מתכנסת ל-A, כנדרש.

 

 

שאלות הבנה:

  1. בחלק הראשון, מדוע כמעט כל אברי הסדרה x_n נמצאים ב-N?
  2. בחלק השני, מדוע ניתן לבחור סדרה אינסופית x_n המתכנסת ל-a כך שהתמונות (f(x_n לא נמצאות ב-I?
  3. מדוע הסדרה המתקבלת (f(x_n אינה מתכנסת ל-A?

Monday, November 16, 2009

Lectures 3-4 (Tue, Nov 17, 24; 9:00-11:00)

Filed under: Rothschild course, lecture — Sergei Yakovenko @ 11:30

Limits of functions and topology of the real line

  • Infinity as the value of the limit: \lim_{n\to\infty} x_n=\pm\infty.
  • Functions of real variable: the domain, range etc (recall). Examples: polynomial and rational functions, \sin x,\ x^\alpha. Compositions: \sin \frac 1x, \ \cos\ln x.
  • 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: f^{-1}(Y\cap Z)=f^{-1}(Y)\cap f^{-1}(Z), the same with \cap. Warning: f(A\cap B)\ne f(A)\cap f(B). Operations on infinitely many sets (unions, intersections).
  • Local and non-local properties of functions.
  • Continuity via open/closed sets.
  • Compactness and its implications.

Saturday, November 14, 2009

אוסף בעיות מספר 1

Filed under: Rothschild course, problems & exercises — galbin @ 7:27

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.

Tuesday, November 10, 2009

Lecture 2 (Thu, Nov 12, 13:30 – 16:30)

Filed under: Rothschild course, lecture — Sergei Yakovenko @ 6:33

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 \mathbb N to \mathbb Q via \mathbb Z. 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 \pi and much, much more
  • Infinite decimal fractions: completion by “adding limits of monotone sequences”.
  • Operations with real numbers: ordered field. Completeness “axiom”.

Wednesday, November 4, 2009

Lecture 1: Nov 5, 2009

Filed under: Rothschild course, lecture — Sergei Yakovenko @ 5:53

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: A=\lim_{n\to\infty}x_n \iff \forall \varepsilon\ \exists N:\ \forall n\ge N\ |x_n-A|<\varepsilon
    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

Caesaria (Rothschild) Programme: Analysis for High School Teachers

Filed under: Rothschild course — Sergei Yakovenko @ 5:34

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.

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