Graphics Reference
In-Depth Information
Greatest lower bound (inf)
When (i)
l
is a lower bound for a set
A
of real numbers, and (ii)
every
m
, greater than
l
, is not a lower bound for
A
,
then and only then,
l
is called the
greatest lower bound
of
A
, and
denoted by inf
A
.
The meanings of the terms 'greatest lower bound' and 'inf' lie in the
words of this definition, and nowhere else!
If a set of numbers has a least member, that number is the greatest
lower bound for the set.
In qn 68 you proved that 0 is the
greatest lower bound
of
x
0
x
1
.
In qn 70 you should havefound that 0 is the
greatest lower bound
of
x
0
x
1
,
x
0
x
1
,
1
(
1)
/
n
n
N
and
q
q
2,
q
Q
.
71 Show that a set
A
cannot have two different greatest lower bounds.
72 If a set
A
has a greatest lower bound inf
A
, and
is any positive
number, why must there be an
a
A
such that
inf
A
a
(inf
A
)
?
73 If
l
is a lower bound for the set
A
, and for any positivenumbr
,
you can find an
a
A
, such that
l
a
l
, must
l
inf
A
?(
Hint
.
Draw a diagram. Suppose
l
m
and
m
is a lower bound for
A
. Try
putting
m
l
.)
74 Let
A
be a set of real numbers with a least upper bound sup
A
, and
let
B
x
x
A
. Provethat inf
B
sup
A
.
75
(i) Prove that the set of numbers
x
x
0
has no greatest
member.
(ii) Prove that the set of numbers
x
x
10
has no greatest
member.
76 List some upper bounds for the set
x
x
0
, and then identify the
set of all upper bounds for
x
x
0
. Does the set of all upper
bounds have a least member?
77 List some upper bounds for the set
x
x
0
, and then identify the
set of all upper bounds for this set. Does the set of upper bounds
have a least member?
78 Let (
a
) be a monotonic decreasing sequence with limit
a
. Prove
that
a
inf
a
n
N
. Useqn 66 and qn 74.