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20 pages in French: Algèbre  
 Définition d'une variable
  
La raison basée sur les  règles et modelés

www.whyslopes.com > Volume 3, Appendices,,  Decimal View of Real Analysis,  199>   B. Bolzano-Weierstrass     Back ] Next ]


Bolzano-Weierstrass Theorem

A set S is said to be infinite if and only if it contains an infinite sequence of points qj (j=1,2,3, ...) with the property that j ¹ k implies qj ¹ qk. A real number A is said to be a limit point of a infinite set of numbers S if and only if every interval I centered at A, no matter how small, must contain infinitely many elements of the set S. Equivalently, a real number A is said to be a limit point of a set S if and only if every interval I centered at A contains at least one element s ¹ A of the set S.

Theorem B.1 [Bolzano-Weierstrass] If an infinite set S is contained in a finite interval [a,b] then the set S has at least one limit point A in the interval [a,b].

 The main idea in the proof given next is as follows. For each whole number k, in the finite interval [a,b] there must be a leftmost subinterval Ik of length 10-k which contains infinitely many points, which in turn must contain an leftmost interval Ik+1 of length [1/2] 10-(k+1) which contains infinitely many points, and so on. It follows that the left end points of these nested (each inside its predecessor) intervals form a Cauchy sequence with a limit L. This limit has the property that in every interval of length [1/2]10-m about L, there are infinitely many points of the infinite set S. Details follow.

Proof of the Bolzano-Weierstrass Theorem.

For each whole number k > 0, the interval [a,b] is covered by subintervals, each of which has length 10-k. We can choose these subintervals so that their end points have a decimal representation that ends k places after the decimal point. If all the subintervals contain only finitely many set members, the set S would be finite. So at least one of the subintervals must have infinitely many elements of S. While we don't have enough information to say which of the finitely many subintervals must has them, there must be at least one subinterval. Choose one, say the leftmost one, and call it Ik.

Now we have a sequence of intervals Ik. The above choice implies that Ik+1 is contained within Ik. The reason for this follows - a proof within a proof: The interval Ik+1 must be to the left, to the right, or inside of the interval Ik. In the first case, Ik+1 would lie in an interval of length 10-k which is to left of Ik and has infinitely points, in particular, those in Ik+1. But the latter is impossible because of the leftmost selection of Ik. In the second case, the interval Ik contains 10 subintervals of length 10-(k+1). But if Ik+1 is to the right of the interval Ik then each of these ten subintervals contains only finitely many points of the set. This contradicts the choice of Ik. The only possibility that remains is that Ik+1 is contained within Ik. That completes this proof within a proof.

The foregoing selection of intervals Ik yields, a nested collection of subintervals, each of length 10-k and each of which contains infinitely many elements of the original set.

End of proof. The above selection process implies that the left end-point of the k-th subinterval has a finite decimal expansion 0.c1c2¼ck which coincides with the first k places of the decimal expansion 0.c1c2¼ckck+1 of the left-end of the next subinterval. This process yields an infinite decimal expansion 0.c1c2c3¼. This defines a real number L = 0.c1c2c3¼ with the property that each neighborhood interval [L-10-k,L+10k] contains infinitely many elements of the original set.

The conclusion (or assumption) that a bounded infinite set of real numbers has a limit point has many consequences in arithmetic-based mathematics. The above proof identifies the leftmost limit point of the set S.

Here for the sake of contradiction, suppose B < A is another limit point of the set S. Then A-B > 10·10-k for some whole number k. Further for such a whole number k, all intervals of length 2·10-m < 10-k centered at B would also contain a point of S. And the latter would be imply there was an interval Jk of length 10-k containing B and to the left of A with infinitely many points of S. But the decimal expansion of A to k-decimal places, say Ak = c1c2¼cp.a1a2¼ak, has the property that the interval Ik = [Ak,Ak+10-k] is the leftmost interval with infinitely many points of S. Yet Jk is to the left of Ik. This is a contradiction. Thus the supposition that there exists a limit point B of S with B < A must be false.
Note the above arguments or reasoning depends on the assumption that an infinite decimal expansions yields a real number. Base two or any other base m ³ 2 could have been used instead. The selection of base ten in the above argument is a historical and cultural preference.
FOOTNOTE: The set-theoretic formulation of modern math moves away from this preference.

The above argument, that is proof, relies on the in principle ability to choose subintervals, one inside another, repeatedly, one for each integer k ³ 1. The above demonstration (proof) is appealing, but some could object to the use in-principle part of this argument. There is no practical or constructive way to make the choice. That is, the choice is possible in principle, but not in practice. For each whole number k, the information that the set S is infinite is insufficient by itself to identify in practice the leftmost interval of length 10-k with an infinite number of points in S.

The above construction of a nested sequence of intervals is a plausible argument for some, but only a figment of the imagination for others. There is a division among mathematicians on whether or not thought-based but impractical choice-based existence arguments (or constructions) are acceptable. The most rigorous and also the most limited perspective is that the above kind of argument is heuristic and somewhat plausible, but not reliable. Another perspective is that the above argument is acceptable and reliable. Suffice it to say that direct, not just in principle, existence proofs are more welcome and more certain in the mathematical reasoning process than other kinds of proof. However, some of the following theorems depend on the above theorem and hence the above in-principle construction.

FOOTNOTE: More Food for thought: Compare and contrast the role of choice in the above argument with the role of Maxwell's Demon in improbable gas dynamics. There may be a limit to what is acceptable in principle as a conclusion-reaching method.
 

the Real Analysis appendices of

Why Slopes
and
More Math

Volume 3

Printed in Canada
ISBN 0-9697564-3-7

These  Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson

Section Entrance
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

Proofs of  one-sided theorems could be of interest in the study of 2D topology.

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/

||
 -/[]\- 
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   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.
Why Slopes & More.Math 1995
Tour their 
forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing Preparing for Scienceends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations
 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5
wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter   
 

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s

3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory
6.    Solving Linear Equations  
Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19
Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21.
Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
-
Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.


 


www.whyslopes.com > Volume 3, Appendices,,  Decimal View of Real Analysis   >   B. Bolzano-Weierstrass     Back ] Next ]


Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep

A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition
   (8 lessons)
Comparison to Subtraction
  (9 lessons)
Multiplication
( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples.
 
-Prime Factorization Aids 
(Learn how to find factors quickly)
-Prime Factorization Examples
 
-Counting & Generating. Factors

-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences
  
Units in calculations
Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way
 & with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for AlgebraSecond, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for AlgebraWhere does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on
Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: Operations on Polynomials:
Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula  The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills. 
Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check

Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in 
Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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