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Solutions Homework 9.
(1) Let
c
n
>
0 in
R
. Prove that
lim
c
n
+1
c
n
≤
lim
n
√
c
n
≤
lim
n
√
c
n
≤
lim
c
n
+1
c
n
.
Solution:
Let
A
=
lim
c
n
+1
c
n
and let
± >
0. If
A
=
∞
, then there is nothing to prove,
so assume
A <
∞
. Then there exists
N
such that
c
n
+1
c
n
< A
+
±
for all
n
≥
N
. This
implies that
c
N
+
m
<
(
A
+
±
)
c
N
+
m

1
<
···
<
(
A
+
±
)
m
c
N
for all
m
≥
1. From this we
conclude that for
n
=
N
+
m
we have
n
√
c
n
<
(
A
+
±
)
±
c
N
(
A
+
±
)
N
²
1
n
.
This implies
lim
n
√
c
n
≤
A
+
±
for all
± >
0 and it follows that
lim
n
√
c
n
≤
lim
c
n
+1
c
n
. The
left hand side of the inequality can be proved similarly, or can be deduced from the
right hand side, if we apply the right hand side inequality to the sequence
d
n
=
1
c
n
and use that
lim
n
√
d
n
=
1
lim
n
√
c
n
.
(2) Let
a
n
≥
0 and
b
n
≥
0. Assume that both (
a
n
) and (
b
n
) are bounded sequences.
(a) Prove that
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 Fall '11
 Schep

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