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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A Finite Covering Theorem
Theorem F.4 [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 x1,x2,¼,xn
in [a,b] with the property that if x is a number in the
interval [a,b] then there exists a j, with 1 £
j £ n and |x-xj|
< g(xj).
| Proof: We can define a sequence wm of
points in [a,b] recursively as follows. First, let w1
= a. Now given wm put wm+1
= min(b,wm+g(wm)).
This defines an non-decreasing sequence wj with
a £ wj £
wj+1 £ b.
Here equality is only possible if wj = b
for some whole number j. Let w = lim j->
¥
wj. Then a £
w £ b.
Suppose w < b. Then for all whole numbers j ³
1, the inequality wj < wj+1
= wj+g(wj) £
w < b holds. Now the continuity of g at all
points x in the interval [a,b] implies g(w)
= limj-> ¥g(wj)
= limj-> ¥ wj+1-wj
= w -w = 0. The foregoing gives
the implication: IF w < b THEN g(w) = 0.
But g(w) > 0 and w £
b. Therefore w = b.
Now there exists an smallest N > 0 such that n ³
N implies b = w ³ wn
> w-g(w) = b-g(b).
The conclusion is satisfied with by putting xj
= wj for 1 £ j
£ N-1 and
setting xN = b. |
The above theorem is needed for the proof of the following assertion.
Question for mathematical adepts: Generalize this proof (using the
lexicographic ordering of points in Rn, so that it
becomes a proof of the equicontinuity of a function f(x1,¼,xn)
continuous on an bounded, closed set S in Rn.
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