MAT 371
Advanced Calculus
/120
March 9, 2005
Test 2
name
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1. a.
Define
limit point
(of a set) and
limit
(of a function).
b.
Define
open set
and
closed set
.
c.
State the
ε

δ
definitions of
continuous
and
uniformly continuous
2. a.
Without proof, find all limit points for each of the following sets (as subsets of
R
).
(i)
Z
.
(ii)
Q
.
(ii) [0
,
1).
(iv)
{
(

1)
n
(1+
n
)
n
:
n
∈
Z
+
}
.
Bonus
:
{
sin
n
:
n
∈
Z
+
}
.
b.
Working from the definition, prove that every constant sequence converges.
c.
Prove that the sequence (
a
n
)
∞
n
=1
defined by
a
n
= (

1)
n n
+1
n
does not converge.
3.
Suppose that
M
∈
R
and (
a
n
)
∞
n
=1
is a sequence in
R
that converges to
L
∈
R
.
Prove that if for all
n
,
a
n
< M
, then
L
≤
M
.
4. a.
Suppose
F
⊆
X
is a closed set in a metric space
X
.
Prove that the complement
O
=
{
x
∈
X
:
x
∈
F
}
of
F
is an open set.
b.
Suppose that
{
F
α
:
α
∈
Λ
}
is a collection of closed subsets of a metric space
X
.
Prove that the intersection
α
∈
Λ
F
α
is a closed subset of
X
.
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 Spring '07
 thieme
 Calculus, Topology, Metric space, Topological space, M R, metric space X., space X. Prove

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