Graphics Reference
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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
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