±
18.100B,
Fall
2002,
Homework
9
Due
by
Noon,
Tuesday
November
26
Rudin:
(1)
Chapter
6,
Problem
12
Proof.
Suppose
that
f
∈R
(
α
)
,
let
C>
0
be
such
that

f
(
x
)
≤
C
for
all
x
∈
[
a,
b
]
.
Given
±>
0
there
exists
a
partition
P
of
[
a,
b
]suchthat
n
(1)
U
(
f,
α,
P
)
−
L
(
f,
α,
P
)=
(
α
(
x
i
)
−
α
(
x
i
−
1
))(
M
i
−
m
i
)
<±
2
/
2
C
i
=1
where
M
i
and
m
i
are
the
supremum
and
inﬁmum
of
f
over
[
x
i
−
1
,x
i
]
.
Con
sider
the
function
given
in
the
hint:
t
−
x
i
−
1
(2)
g
(
t
)=
x
i
−
t
f
(
x
i
−
1
)+
f
(
x
i
)
,t
∈
[
x
i
−
1
,x
i
]
.
x
i
−
x
i
−
1
x
i
−
x
i
−
1
Note
that
the
value
at
t
=
x
i
is
independent
of
choice
even
if
there
are
two
intervals
of
which
it
is
an
end
point.
On
[
x
i
−
1
,x
i
]
,g
is
continuous
since
it
is
linear
there
and
it
is
continuous
at
each
x
i
,
hence
is
continuous
everywhere.
On
[
x
i
−
1
,x
i
]
,g
takes
values
in
[
m
i
,M
i
]
since
its
maximum
and
minimum
occur
at
the
ends
(it
is
linear)
and
these
are
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 Fall '09
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