Graphics Reference
In-Depth Information
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
.