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Lebl-Basic-Analysis-I
· Problem 6.2.10
Sequences of Functions
Mathematics · dinis-theorem
Problem
(Hard) Prove Dini's theorem: Let
f
n
:
[
a
,
b
]
→
R
f_n : [a,b] \to \mathbb{R}
f
n
:
[
a
,
b
]
→
R
be a sequence of continuous functions such that
0
≤
f
n
+
1
(
x
)
≤
f
n
(
x
)
≤
⋯
≤
f
1
(
x
)
for all
n
∈
N
.
0 \leq f_{n+1}(x) \leq f_n(x) \leq \cdots \leq f_1(x) \quad \text{for all } n \in \mathbb{N}.
0
≤
f
n
+
1
(
x
)
≤
f
n
(
x
)
≤
⋯
≤
f
1
(
x
)
for all
n
∈
N
.
Suppose
{
f
n
}
n
=
1
∞
\{f_n\}_{n=1}^\infty
{
f
n
}
n
=
1
∞
converges pointwise to
0
0
0
. Show that
{
f
n
}
n
=
1
∞
\{f_n\}_{n=1}^\infty
{
f
n
}
n
=
1
∞
converges to zero uniformly.
Related theory
The Prime Number Theorem And Dirichlet S Theorem
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