Homework 3 – Math CS 120, Winter 2009
Due on Thursday, January 29th, 2009
1. Consider a sequence of real numbers
{
x
n
}
n
∈
N
⊂
R
. Prove that if there
exists a subsequence of
{
x
n
}
n
∈
N
that converges to
b
∈
R
, then
lim inf
n
→∞
x
n
≤
b
≤
lim sup
n
→∞
x
n
.
2. Find
(a)
lim
x
→
0
√
1 +
x

√
1

x
x
.
(b)
lim
x
→
1
x
5

1
x
4

1
.
(c)
lim
x
→
2
3
√
x

3
√
2
√
x

√
2
.
3. Let
f
: [
a, b
]
→
R
be a continuous function, and consider
x
1
, x
2
∈
[
a, b
],
x
1
≤
x
2
,
y
∈
R
such that
f
(
x
1
)
≤
y
≤
f
(
x
2
)
.
Prove that there exists
c
∈
[
x
1
, x
2
] such that
f
(
c
) =
y
.
4. A point
a
∈
R
n
is called a boundary point of the set
A
⊂
R
n
if every
open ball centered at
a
intersects both
A
and
A
c
=
R
n
\
A
.
The
boundary
of the set
A
is the set of all its boundary points. Prove that
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 Fall '09
 Topology, Real Numbers, lim, Topological space, Closed set, lim sup xn, lim inf xn

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