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Mathematics 250
Fall 2008
Proof Practice # 3
For a continuous realvalued function
f
deﬁned on an interval [
a,b
] in
R
, and a partition
P
of [
a,b
],
let
R
P
(
f
) denote the lower Riemann sum for
f
and
P
, obtained by evaluating
f
at its minimum point in
each subinterval deﬁned by
P
. Similarly, let
R
P
(
f
) denote the upper Riemann sum for
f
and
P
. Let
Q
be
another partition, and suppose that
Q
is a reﬁnement of
P
. Show that
R
P
(
f
)
≤
R
Q
(
f
)
≤
R
Q
(
f
)
≤
R
Q
(
f
)
.
(
Mon
)
Response: A main issue with writing a reasonable argument for this statement is managing the situation
so that the notation doesn’t become overwhelming. One strategy for doing this is to think of a special case
that is simple but representative. In this situation, such a special case can be found.
We recall that a partition
Q
reﬁnes a partition
P
if every subdivision point of
P
also belongs to
Q
. In
other words,
Q
can be obtained from
P
by adding more points. This suggests the idea of adding the points
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This note was uploaded on 09/24/2009 for the course MATH 250 taught by Professor Rogerhowe during the Fall '06 term at Yale.
 Fall '06
 RogerHowe
 Math

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