Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Lebl-Basic-Analysis-I
· Problem 5.1.10
The Riemann Integral
Mathematics · darboux-sums
Problem
Let
f
:
[
0
,
1
]
→
R
f : [0,1] \to \mathbb{R}
f
:
[
0
,
1
]
→
R
be a bounded function. Let
P
n
=
{
x
0
,
x
1
,
…
,
x
n
}
P_n = \{x_0, x_1, \ldots, x_n\}
P
n
=
{
x
0
,
x
1
,
…
,
x
n
}
be a uniform partition of
[
0
,
1
]
[0,1]
[
0
,
1
]
, that is,
x
i
=
i
/
n
x_i = i/n
x
i
=
i
/
n
. Is
{
L
(
P
n
,
f
)
}
n
=
1
∞
\bigl\{L(P_n, f)\bigr\}_{n=1}^\infty
{
L
(
P
n
,
f
)
}
n
=
1
∞
always monotone? Yes/No: Prove or find a counterexample.
Related theory
Refinement And Monotonicity Of Darboux Sums
Browse the problem library