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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Greatest and Least Bounds
A real number M is said to be an upper bound for a set S
of real numbers if M ³ s whenever s
belongs to the set S. Similarly, a real number K is said to be a lower
bound for a set S of real numbers if K £
s whenever s belongs to the set S. The following theorem
asserts the existence of a least upper bound B = sup(S) and
also a greatest lower bound A = inf(S) for each set S
contained in a finite interval [a,b].
Theorem D.1 [On Greatest and Least Bounds] If a set S is
contained in a finite interval [a,b] then there is a subinterval [A,B]
containing S such that among all intervals of the form [p,q]
containing S, the interval [A,B] is the smallest. The
number A in this theorem is called the greatest lower bound (The name
most positive lower bound might be more appropriate) of the set S. It can
be obtained to k decimal places by modifying the proof of the
Bolzano-Weierstrass theorem, so that the Ik is the
leftmost interval of length 10-k
with the property that there are no points of S to the left of it. The
left-end point of Ik then yields A to k
decimal places. The decimal expansion defines a number A = inf(S)
(read the infimum of S) with the property that no elements of S is
to the left of it. In other words, the number A = inf(S) is the
greatest number with the property that no numbers in S are lower than it.
It is called the greatest lower bound. That is, the number A = inf(S)
is lower or £ all numbers in the set S, and
it is the largest number with this property. Moreover, by construction, in every
interval of the form [A, A+10-k]
where k is a whole number, there must be at least one element of S.
Thus, if A is not an element of S, then the set S must be
infinite and A is the leftmost limit point of S. On the other
hand, if A belongs to S, it is the leftmost or least element of S
and S could be finite or infinite. (Exercise: What can be said about the
membership of A in the set S when the set S is finite?)
Similarly, the supremum of S is the number sup(S) = B.
It is also called the least upper bound (The name least positive upper bound
might be more appropriate) of the set S. It can be obtained to k
decimal places by modifying the second (or first proof) so that Ik
is the rightmost interval of length 10-k
with the property that there are no points of S to the right of it. The
left-end point of Ik then yields B to k
decimal places. The number B = sup(S) is the leftmost point or
number with the property that no element of S is to the right of it. In
other words, the number B is the least number with the property that no
numbers in S are above it. Here B is above or ³
all numbers in the set S. Moreover, by construction, in every interval of
the form [B-10-k,B]
where k is a whole number, there must be at least one element of S.
Thus if B is not an element of S, then the set S must be
infinite and B is the leftmost limit point of S. On the other
hand, if B belongs to S, it is the rightmost or greatest element
of S. Here S may be finite or infinite.
Question. What can be said about the existence of the
least and greatest lower bounds of a set S when the latter is contained
in a semi-infinite interval of the form [a,¥)
or (-¥,b]?
Two Exercises
- What can be said about the membership of B in the set S if
the set S is finite?
- Show if S = U ÈV Ì
[a,b] then
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sup
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(S) = |
max
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sup
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(U), |
sup
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(V)) |
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and
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inf
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(S) = |
min
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( |
inf
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(U), |
inf
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(V)) |
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