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57 Suppose n N a
a
1, then the sequence is bounded above
by max( a
, a
,..., a
, a
1) and bounded below by
min( a
, a
,..., a
, a
1). Now ( a
) is a Cauchy sequence, so given
0, there is an N
and an N
such that n
N
a
a
,
and n
N
a
a , so for any
a .
58 With the given inequalities, all the later terms of the sequence lie
between a
n
max( N
, N
), n n
a
and a
,so a
a
1/ n , and the sequence is a
Cauchy sequence.
59 The greatest member is 4. Any finite set of numbers, which is not
empty, has a greatest member. 1.
60
(i) (a) Yes. (b) No.
(ii) (a) Yes. (b) No.
61 UB
2 except for (v), (vi) and (viii).
62 Not for 61(vi) or (viii). Is for 61(v).
63 Suppose u and v are least upper bounds for A , and v u , then v is
not a least upper bound by definition (ii).
64 If not, then (sup A ) is an upper bound and so sup A is not least.
So there is an a
(sup A
1, sup A ] with a
A . But if
sup A
A , a
(sup A
1, sup A ), and then there is an a
( a
, sup A )
with a
A , and an a
( a
, sup A ), etc.
65 Now suppose there is an upper bound v with v u . Taking
u v ,
there is an a such that u
a
u , and thus v
a
u .So v is not
an upper bound and u
sup A .
66 Since( a
)
a , for any
0,
a
a
for suMciently large n ,or
a a
a
. From qn 3.80 (b), a
a for all n ,so a is an upper
bound for
a
n
N
and a
a
a , for suMciently large n .Byqn
65, a
sup
a
n
N
.
68 (a) Yes, (b) no.
69
x 0 x 1 .
70 Can find lower bound
0 for (v), (vi), (viii).
71 If there were two, then the greater of the two would not be a lower
bound.
72 If not, inf A
would bea lowr bound.
73 If l is not the greatest lower bound, then there is a lower bound m with
l
m . Now let
m
l , and there is an a such that
l
a
l
m ,so m is not a lower bound and l
inf A .
 
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