Math 1432
Notes – Week 9
10.1 % THE LEAST UPPER BOUND AXIOM
M
is an
upper bound
for
S
if
x
≤
M
for all
x
ε
S.
The
least upper bound
of
S
is an upper bound that is less than or equal to any other upper bound
for
S.
LEAST UPPER BOUND AXIOM
Every nonempty set of real numbers that has an upper bound has a
least
upper bound.
Examples:
1.
{1,2,3,4}
S
=
2. [?4, 2]
3.
(
)
,8
−∞
4.
(
)
5,
∞
5.
2
{ :
16}
S
x x
=
≤
6.
1 1 1 1
1
1,
,
,
,
,
,
,
2 3 4 5
1000
S
⎧
⎫
=
⎨
⎬
⎩
⎭
L
L
THEOREM 10.1.2
If
M
is the least upper bound of the set
S
and ε is a positive number, then there is at least one
number
s
in
S
such that
M
−ε
< s
≤
M
.
Example:
1 2 3 4
,
,
,
,
0.01
2 3 4 5
S
ε
⎧
⎫
=
=
⎨
⎬
⎩
⎭
L
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THEOREM 10.1.3
Every nonempty set of real numbers that has a lower bound has a
greatest
lower bound.
Examples:
1.
{1,2,3,4}
S
=
2. [?4, 2]
3.
(
)
,8
−∞
4.
(
)
5,
∞
5.
2
{ :
16}
S
x x
=
≤
6.
1 1 1 1
1
1,
,
,
,
,
,
,
2 3 4 5
1000
S
⎧
⎫
=
⎨
⎬
⎩
⎭
L
L
THEOREM 10.1.4
If
m
is the greatest lower bound of the set
S
and ε is a positive number, then there is at least one
number
s
in
S
such that
m
≤
s < m
+ ε.
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 Spring '08
 morgan
 Math, Calculus, lim, Order theory, Zagreb, Monotonic function

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