Graphics Reference
In-Depth Information
(
Hint
. Draw a diagram. Suppose
v
u
and that
v
is also an upper
bound for
A
. Try putting
u
v
.)
66 Let (
a
) be a monotonic increasing sequence with limit
a
. Provethat
a
a
n
N
sup
. Useqn 3.80.
Lower bounds and least members
67 Give some examples of sets of numbers which have a least member.
Make sure some of them are finite and some are infinite. See Fig.
4.67.
68
?
(b) Can any number greater than 0 be a lower bound for
(a) Is 0 a lower bound for
x
0
x
1
x
0
x
1
?
69 Give an example of a bounded set of numbers which does not have
a least member.
70 Check that 0 is a lower bound for each of the following sets:
(i)
x
0
x
1
,
(ii)
x
0
x
1
,
(iii)
x
0
x
,
x
1, 1
x
1
,
,1
x
1
,
(v)
1
1/
n
n
N
,
(vi)
2
1/
n
n
N
,
(vii)
1
(
1)
/
n
n
N
,
(viii)
x
x
2,
x
Q
,
(ix)
(iv)
x
0
x
q
q
2,
q
Q
.
For which of these sets can you find a number greater than 0 which
is still a lower bound for the set?
inf
inf
inf
Fig 4.67
