Graphics Reference
In-Depth Information
74 For all
a
,
a
sup
A
sup
A
a
, and
sup
A
is a lower bound
for
B
.
For any
0, there is an
a
A
such that sup
A
a
sup
A
,
so that
sup
A
a
sup
A
, and by qn 73, inf
B
sup
A
.
75
(i) If
g
were greatest, then
g
/2 would be greater!
(ii) If
g
were greatest, (
g
10)/2 would be greater!
76 All positive numbers and 0 are upper bounds for the set of negative
numbers. The least is 0.
77 Answer as 76.
78 (
a
) is monotonic increasing with limit
a
,so
a
sup
a
n
N
by qn 66. Now
a
inf
a
n
N
by qn 74.
79 By qn 66, the least upper bound of the terms of an infinite decimal
sequence is the infinite decimal. So if every non-empty set of real
numbers which is bounded above has a least upper bound, then every
infinite decimal sequence is convergent.
80
(i) The sequence (
l
) is monotonic increasing and bounded above by
.
(ii) The sequence (
u
u
) is monotonic decreasing and bounded below by
l
.
(iii)
u
l
(
)
(
u
l
), so by the difference rule, the limits are
equal.
(v) Since
l
is not a upper bound for
X
and
s
is, there is an
x
with
l
x
s
, for every
n
. Now (
l
)
s
, so given
0,
l
s
for su
M
ciently large
n
, and
s
l
x
s
.So
s
sup
X
by qn
65.
81 A non-empty set of real numbers which is bounded below has a
greatest lower bound.
82 inf
A
inf
B
sup
B
sup
A
. Can take
A
x
0
x
1
and
B
x
0
x
1
.
83
) is bounded, qn 46.
(ii) By the closed interval property, qn 3.78.
(iii) By qns 80 and 81.
(i) (
a