the
Real Analysis appendices of
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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These Real Analysis appendices
continue the decimal
viewpoint of limits, continuity and convergence in chapter 14. and
this further lesson
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.
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Appendix B
Pigeon Hole Principle
The Finite Case
The finite pigeon hole principle is given by the contrapositive of the
following implication: If there is no more than one pigeon in each of n
holes, then the n holes contain at most n pigeons. The
equivalent contrapositive of this implication follows. If n holes
contain more than n pigeons, then at least one hole contains more than
one pigeon. Here it is possible to examine each hole or location, one at a
time. After at most n inspections, a first hole with more than one pigeon
will be found.
Instead of pigeons and holes, we could speak of elements and
sets in accordance with the set-theoretic formulation favored in the axiomatic
formulation and codification of modern mathematics.
The Infinite Case
More generally, the infinite pigeon hole principle is as follows. If n
holes contains an infinite number of points (small pigeons) then at least one of
the holes contain an infinite number. In particular, if the holes are
labeled or ordered from 1 to n, then there must be a first hole with infinitely
many points (small pigeons) in it. But a finite number of inspections need not
say which one. The infinite pigeon hole principle is the contrapositive of the
following implication rule: If each of n holes contains finitely many
pigeons then all n holes together contain finitely many pigeons.
Imagine for instance that we try to inspect the holes one at a time in
sequence in the hope of identifying the first hole with infinitely many points.
Further suppose that the points in a hole can be counted one per second. Now if
a hole has finitely many points, we can count them all in a finite time. But if
a hole has infinitely many points, counting them one per second will begin but
never end. Even after counting a large number of points in the hole, there still
may be a small, large or infinite number of points in the hole. So even if a
count does not appear to be ending, we cannot say that it does.
Food for thought: In principle perhaps, time and temporal notions have
no place in the comprehension or visualization of mathematics: we can see or
understand everything at once. But logical and numerical reasoning or
computations as they are followed or done depend on a sequence of operations
or counts in time. In particular, mathematical reasoning takes time to be done
or followed. Yet with hindsight, that reasoning and the results obtained can
be reviewed and understood with a fleeting thought - divorced from the passage
of time. The reasoning and the results appear to have an objective (Platonic)
existence apart from the passage of time. To say more would delve into several
philosophies of mathematics and logic.
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