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Introduction to Analysis: Fall 2008
Practice problems V
MTH 4101/5101
10/21/2008
1. Show that the sequence
{
1
(
n
2
+1)
}
converges to 0.
Solution:
Let
± >
0 be given. For
n
∈
IN
, we have
1
n
2
+ 1
<
1
n
2
≤
1
n
.
Choose
N
such that
1
N
< ±
. Then we have,

1
n
2
+1

0

<
1
n
< ±.
2. Show that the sequence
{
3
n
+2
(
n
+1)
}
converges to 3.
Solution:
Let
± >
0 be given. We
±
±
±
±
3
n
+ 2
n
+ 1

3
±
±
±
±
=
±
±
±
±
3
n
+ 2

3
n

3
n
+ 1
±
±
±
±
=
±
±
±
±

1
n
+ 1
±
±
±
±
=
1
n
+ 1
<
1
n
.
3. If 0
< b <
1 show that lim
n
→∞
b
n
= 0.
Solution;
If
± >
0 is given, we have
b
n
< ±
⇔
n
ln
b <
ln
±
⇔
n >
ln
±
ln
b
.
(The last inequality is reversed because ln
b <
0.)
4. Let
{
x
n
}
be a sequence of real numbers and let
x
∈
IR.
If
{
a
n
}
is a sequence of
positive real numbers, with lim
n
→∞
a
n
= 0 and if for some constant
C >
0 and
some
m
∈
IN
we have

x
n

x
 ≤
Ca
n
for all
n
≥
m
show that lim
x
n
=
x
.
Solution:
Let
±.
0 be given. Since lim
a
n
= 0 we know that there exists
K
such that such that
n
≥
K
implies
a
n
=

a
n

0

<
±
C
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 Spring '10
 SHALOM
 Linear Algebra, Algebra

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