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
|
sup
|
(S) = |
max
|
( |
sup
|
(U), |
sup
|
(V)) |
|
and
|
inf
|
(S) = |
min
|
( |
inf
|
(U), |
inf
|
(V)) |
|
| |
the Real Analysis appendices of
Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
Presenting Appendices from Volume 3, Why
Slopes and More Math, If the epsilon-delta viewpoint of
limits, continuity and convergence is not yet comfortable, see Chapters 14
to 19 in Volume 3 are related.
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.
Vol 1A Logic Postscripts
online only include
Proof
by Absurdity alias proof by contradiction
How
the demand for consistency supports the law of the excluded middle
Reality
versus or with the aid of Imagination
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