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Math 5615H. October 5, 2011.
Midterm Exam 1. Problems and Solutions.
Problem 1.
(10 points). Let
A
and
B
be nonempty bounded subsets of
R
. Show that
sup(
A
∪
B
) = sup
{
sup
A,
sup
B
}
.
Proof.
Denote
M
1
:= sup(
A
∪
B
)
, M
2
:= sup
{
sup
A,
sup
B
}
. We have
x
≤
sup
A
≤
M
2
∀
x
∈
A,
and
x
≤
sup
B
≤
M
2
∀
x
∈
B.
Hence
x
≤
M
2
∀
x
∈
A
∪
B
, which implies
M
1
≤
M
2
.
On the other hand, since
A
⊆
A
∪
B
and
B
⊆
A
∪
B
, we also have sup
A
≤
M
1
,
sup
B
≤
M
1
,
so that
M
2
≤
M
1
. Together with the opposite inequality
M
1
≤
M
2
, this yields
M
1
=
M
2
.
Problem 2.
(10 points). Find an
explicit
expression for a
onetoone
mapping
f
of
A
onto
B
,
where
A
:= [0
,
1] =
{
x
∈
R
: 0
≤
x
≤
1
}
,
B
:= [0
,
1) =
{
x
∈
R
: 0
≤
x <
1
}
.
Hint:
First deﬁne
f
on the sequence
A
0
:=
{
a
n
= 1
/n
;
n
= 1
,
2
,
3
,...
}
.
Solution.
One can take
f
(
x
) =
(
a
n
+1
if
x
=
a
n
∈
A
0
,
x
if
x /
∈
A
0
.
.
Problem 3.
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