1. Cauchy, but not monotone.
2. Monotone, but not Cauchy.
3. Bounded, but not Cauchy.
Solutions:
1. (
(
−
10
n
n
) is Cauchy, but not monotone.
2. (1
,
2
,
3
,
4
, ...
) is monotone, but not Cauchy.
3. (1
,
0
,
1
,
0
,
1
,
0
, ...
) is bounded, but not Cauchy.
�
•
4.3-6* Let
(
a
n
)
and
(
b
n
)
be monotone sequences. Prove or give a counterex-
ample,
1. The sequence
(
c
n
)
given by
c
n
=
ka
n
is monotone for any
k
∈
R
.
2. The sequence
(
c
n
)
given by
c
n
=
a
n
/b
n
is monotone, where
b
n
�
= 0
for all
n
∈
N
.
Solutions:
1. Yes. (
c
n
) = (
ka
n
) is monotone. In fact, if
k >
0, we have that
ka
n
≤
ka
n
+1
if
and only if
a
n
≤
a
n
+1
; and that
ka
n
≥
ka
n
+1
if and only if
a
n
≥
a
n
+1
. Therefore,
(
ka
n
) is monotone if and only if (
a
n
) is monotone. Similarly we can consider the
case
k <
0 and
k
= 0.
2. No. Here is a counterexample: (
a
n
) = (1
,
2
,
2
,
3
,
3
,
4
,
4
,
....
) and (
b
n
) = (1
,
1
,
2
,
2,
3
,
3
,
4
,
4
, ...
) are monotone. But the sequence
�
a
n
b
n
�
=
�
1
,
2
,
1
,
3
2
,
1
,
4
3
, ...
�
is not monotone.
•
4.3-9* Suppose
x >
0
. De
fi
ne a seqeunce
(
s
n
)
by
s
1
=
k
and
s
n
+1
=
s
2
n
+
x
2
s
n
for
n
∈
N
. Prove that for any
k >
0
,
lim
s
n
=
√
x
.
Solutions:
(
s
n
) is bounded below
We claim that
s
n
≥
√
x
for all
n
≥
2, i.e.,
s
2
n
−
x
≥
0 for all
n
≥
2. In fact,
21