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.

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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