3 4 find an example of a sequence of real numbers

4.3-4* Find an example of a sequence of real numbers satisfying each set of
properties.
20

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

s
2
k +1
− x =
�
s
2
k
+ x
2 s
k
�
2
− x =
s
4
k
+ 2 s
2
k
x + x
2
4 s
2
k
− x
=
s
4
k
+ 2 s
2
k
x + x
2
− 4 s
2
k
x
4 k 2
=
( s
2
k
− 2)
2
4 k 2
≥ 0 .
holds for any k ≥ 1.
( s
n
) is decreasing : We want to prove that ( s
n
) is decreasing for n ≥ 2: s
n
− s
n +1
≥ 0
for all n ≥ 2. In fact,
s
n
− s
n +1
= s
n
−
x 2
n
+ x
2 s
n
=
s 2
n
− x
2 s
n
≥ 0 .
Here for the last inequality, we have used the ineqyality s
2
n
− x ≥ 0 proved above.