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Lebl-Basic-Analysis-I
· Problem 5.1.3
The Riemann Integral
Mathematics · riemann-integrability
Problem
Let
f
:
[
a
,
b
]
→
R
f : [a,b] \to \mathbb{R}
f
:
[
a
,
b
]
→
R
be a bounded function. Suppose there exists a sequence of partitions
{
P
k
}
k
=
1
∞
\{P_k\}_{k=1}^\infty
{
P
k
}
k
=
1
∞
of
[
a
,
b
]
[a,b]
[
a
,
b
]
such that
lim
k
→
∞
(
U
(
P
k
,
f
)
−
L
(
P
k
,
f
)
)
=
0.
\lim_{k\to\infty} \bigl(U(P_k, f) - L(P_k, f)\bigr) = 0.
k
→
∞
lim
(
U
(
P
k
,
f
)
−
L
(
P
k
,
f
)
)
=
0.
Show that
f
f
f
is Riemann integrable and that
∫
a
b
f
=
lim
k
→
∞
U
(
P
k
,
f
)
=
lim
k
→
∞
L
(
P
k
,
f
)
.
\int_a^b f = \lim_{k\to\infty} U(P_k, f) = \lim_{k\to\infty} L(P_k, f).
∫
a
b
f
=
k
→
∞
lim
U
(
P
k
,
f
)
=
k
→
∞
lim
L
(
P
k
,
f
)
.
Related theory
Lebesgue S Criterion For Riemann Integrability
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