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Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
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for college students, gifted teens, home-tutoring and K1-12 schooling, with chapters
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and Pattern
Based Reason to inform and amuse thinkers and avid readers, studying or not. Enjoy.
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Logic mastery strengthens comprehension and improve home,
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About: Site material shows how common troubles
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Teachers & Tutors: This December 2011, 5-phase framework
offers a context for mathematics & logic education. Phases 1 to
3 may focus on skills with actual or potential local value for
adult & daily life. College-oriented phases 5 & 4 focus on
calculus & preparation for it. Phases 1 to 4 may also serve
trades & professions not dependent on calculus. Reform: look
before you leap - plan all in detail first.
Site Review: Math resources ... span ... arithmetic, logic,
algebra, calculus, complex numbers, and Euclidean geometry. Lessons
and how-tos .... provide a good foundation for high school and
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Home < Advanced Calculus - Volume 3 Appendices << F.4 Finite Covering Theorem
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A Finite Covering Theorem
Appendices, Volume 3, Why Slopes and More Math.
[A Finite Covering Theorem] Let a < b. At each x in the interval [a,b]
, suppose that the function g(x) is continuous and suppose that g(x) >
0. Then there exists a whole number $n>0$ and a set of numbers
$x_1,x_2,\ldots,x_n$ in $[a,b]$ with the property that if $x$ is a number
in the interval $[a,b]$ then there exists a whole number $j$ with $1\le j
\le n$ such that $|x-x_j| \lt g(x_j)$.
Proof of Finite Covering Theorem.
We can define a sequence $w_m$ of points in $[a,b]$ recursively as
follows. First, let $w_1=a$. Now given $w_m$ put
$w_{m+1}=\min(b,w_m+g(w_m))$. This defines an non-decreasing sequence
$w_j$ with $a\le w_j\le w_{j+1} \le b$. Here equality is only
possible if $w_j=b$ for some whole number $j$.
Let \[w=\lim_{j\to \infty} w_j.\]
Then $a\le w \le b$.
Suppose $w \lt b$. Then for all whole numbers $j\ge 1$, the
inequality $w_j \lt w_{j+1}=w_j+g(w_j)\le w \lt b$ holds. Now the
continuity of $g$ at all points $x$ in the interval $[a,b]$ implies
\[g(w)=\lim_{j\to\infty} g(w_j)=\lim_{j\to\infty} w_{j+1}-w_j =w -w
=0.\] The foregoing gives the implication: IF $w0$
and $w\le b$. Therefore $w=b$.
Now there exists an smallest $N>0$ such that
$n\ge N$ implies $b=w\ge w_n > w-g(w)=b-g(b)$. The conclusion is
satisfied with by putting $x_j=w_j$ for $1\le j \le N-1$ and setting
$x_N=b$.
The above theorem is needed for the proof of the
next theorem.
Exercise for mathematical adepts: Generalize this
proof (using the lexicographic ordering of points in $R^n$, so that it
becomes a proof of the equicontinuity of a function $f(x_1,\ldots,x_n)$
continuous on an bounded, closed set $S$ in $R^n$.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
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protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
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writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
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Is your child able to add, subtract and multiply amounts
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work with maps and plans, and measure length, weight-mass and
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providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
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Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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Home < Advanced Calculus - Volume 3 Appendices << F.4 Finite Covering Theorem
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14][15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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